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Properties

Label 1-967-967.161-r0-0-0
Degree 11
Conductor 967967
Sign 0.998−0.0618i0.998 - 0.0618i
Analytic cond. 4.490724.49072
Root an. cond. 4.490724.49072
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.936 + 0.350i)2-s + (−0.822 − 0.568i)3-s + (0.754 − 0.655i)4-s + (0.672 − 0.739i)5-s + (0.969 + 0.244i)6-s + (0.719 + 0.694i)7-s + (−0.477 + 0.878i)8-s + (0.353 + 0.935i)9-s + (−0.371 + 0.928i)10-s + (−0.511 + 0.859i)11-s + (−0.993 + 0.110i)12-s + (0.999 − 0.0130i)13-s + (−0.917 − 0.398i)14-s + (−0.974 + 0.225i)15-s + (0.139 − 0.990i)16-s + (0.996 + 0.0779i)17-s + ⋯
L(s)  = 1  + (−0.936 + 0.350i)2-s + (−0.822 − 0.568i)3-s + (0.754 − 0.655i)4-s + (0.672 − 0.739i)5-s + (0.969 + 0.244i)6-s + (0.719 + 0.694i)7-s + (−0.477 + 0.878i)8-s + (0.353 + 0.935i)9-s + (−0.371 + 0.928i)10-s + (−0.511 + 0.859i)11-s + (−0.993 + 0.110i)12-s + (0.999 − 0.0130i)13-s + (−0.917 − 0.398i)14-s + (−0.974 + 0.225i)15-s + (0.139 − 0.990i)16-s + (0.996 + 0.0779i)17-s + ⋯

Functional equation

Λ(s)=(967s/2 ΓR(s) L(s)=((0.998−0.0618i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(967s/2 ΓR(s) L(s)=((0.998−0.0618i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 967967
Sign: 0.998−0.0618i0.998 - 0.0618i
Analytic conductor: 4.490724.49072
Root analytic conductor: 4.490724.49072
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ967(161,⋅)\chi_{967} (161, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 967, (0: ), 0.998−0.0618i)(1,\ 967,\ (0:\ ),\ 0.998 - 0.0618i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.9766396099+0.03021989345i0.9766396099 + 0.03021989345i
L(12)L(\frac12) ≈\approx 0.9766396099+0.03021989345i0.9766396099 + 0.03021989345i
L(1)L(1) ≈\approx 0.7444752623+0.02643874580i0.7444752623 + 0.02643874580i
L(1)L(1) ≈\approx 0.7444752623+0.02643874580i0.7444752623 + 0.02643874580i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad967 1 1
good2 1+(−0.936+0.350i)T 1 + (-0.936 + 0.350i)T
3 1+(−0.822−0.568i)T 1 + (-0.822 - 0.568i)T
5 1+(0.672−0.739i)T 1 + (0.672 - 0.739i)T
7 1+(0.719+0.694i)T 1 + (0.719 + 0.694i)T
11 1+(−0.511+0.859i)T 1 + (-0.511 + 0.859i)T
13 1+(0.999−0.0130i)T 1 + (0.999 - 0.0130i)T
17 1+(0.996+0.0779i)T 1 + (0.996 + 0.0779i)T
19 1+(0.505−0.862i)T 1 + (0.505 - 0.862i)T
23 1+(0.527−0.849i)T 1 + (0.527 - 0.849i)T
29 1+(−0.724+0.689i)T 1 + (-0.724 + 0.689i)T
31 1+(−0.992−0.123i)T 1 + (-0.992 - 0.123i)T
37 1+(−0.895+0.445i)T 1 + (-0.895 + 0.445i)T
41 1+(0.854+0.519i)T 1 + (0.854 + 0.519i)T
43 1+(0.929−0.368i)T 1 + (0.929 - 0.368i)T
47 1+(0.975+0.219i)T 1 + (0.975 + 0.219i)T
53 1+(−0.648+0.761i)T 1 + (-0.648 + 0.761i)T
59 1+(0.914+0.404i)T 1 + (0.914 + 0.404i)T
61 1+(−0.877+0.480i)T 1 + (-0.877 + 0.480i)T
67 1+(0.389−0.921i)T 1 + (0.389 - 0.921i)T
71 1+(0.165+0.986i)T 1 + (0.165 + 0.986i)T
73 1+(−0.648−0.761i)T 1 + (-0.648 - 0.761i)T
79 1+(−0.597+0.801i)T 1 + (-0.597 + 0.801i)T
83 1+(0.795−0.605i)T 1 + (0.795 - 0.605i)T
89 1+(0.602+0.797i)T 1 + (0.602 + 0.797i)T
97 1+(−0.222+0.974i)T 1 + (-0.222 + 0.974i)T
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   L(s)=∏p (1−αp p−s)−1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−21.389655417663376537567348925190, −21.02349589635418591127922357931, −20.524118801094243377674524257719, −19.01703042836888036169427850769, −18.554734379911133340578053220236, −17.754120280433970463229006689160, −17.228640270311657908385768252885, −16.39790136069432619720908403306, −15.80600474783535268226491459227, −14.70031970633648180099362444719, −13.82834531760720153914053469511, −12.8158089817734781055769158072, −11.62792557524111328710366228603, −11.017089133799139244089141732349, −10.59712299034742002271809447215, −9.82634575732517643632340955568, −8.98855481971054420017956977999, −7.79780978177539770741197954383, −7.14557994627257619971343700764, −5.931706138064847248210035961614, −5.50284599035014857557891874320, −3.789284843993496647143510917, −3.31207200380647999940853854439, −1.77976149816798144041614853813, −0.8793168137405094711001349383, 0.97358061051806961634206968327, 1.673252946227874494839981057760, 2.566114696245051236996753859300, 4.71692794691585390998014136272, 5.447709335008727922737713217431, 5.94349606265658823023510525105, 7.07862043250064244809888235108, 7.81610089530430815703876172828, 8.73255517666386487132056135014, 9.417157005929789104351607538276, 10.535346029418221683454925853829, 11.08591470357090328530741641763, 12.1455265636888864069492790108, 12.69234824695540987745069595085, 13.76899086159170418677694523201, 14.7613167949666286523016076731, 15.75343058554824114897798444332, 16.39784671014585700860175543666, 17.22028471023276489077695682596, 17.82024058968335718455475254567, 18.37366094487234243498992499731, 18.944318192627081574640431598725, 20.27492222412381386200865363762, 20.76644823733028444934990954609, 21.59165888119056236537891266434

Graph of the ZZ-function along the critical line