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Properties

Label 1-967-967.16-r0-0-0
Degree 11
Conductor 967967
Sign −0.492+0.870i-0.492 + 0.870i
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  + (0.328 + 0.944i)2-s + (0.981 + 0.193i)3-s + (−0.783 + 0.620i)4-s + (−0.953 − 0.300i)5-s + (0.139 + 0.990i)6-s + (0.0617 − 0.998i)7-s + (−0.844 − 0.536i)8-s + (0.924 + 0.380i)9-s + (−0.0292 − 0.999i)10-s + (0.279 + 0.960i)11-s + (−0.889 + 0.457i)12-s + (−0.971 + 0.238i)13-s + (0.962 − 0.269i)14-s + (−0.877 − 0.480i)15-s + (0.228 − 0.973i)16-s + (0.126 + 0.991i)17-s + ⋯
L(s)  = 1  + (0.328 + 0.944i)2-s + (0.981 + 0.193i)3-s + (−0.783 + 0.620i)4-s + (−0.953 − 0.300i)5-s + (0.139 + 0.990i)6-s + (0.0617 − 0.998i)7-s + (−0.844 − 0.536i)8-s + (0.924 + 0.380i)9-s + (−0.0292 − 0.999i)10-s + (0.279 + 0.960i)11-s + (−0.889 + 0.457i)12-s + (−0.971 + 0.238i)13-s + (0.962 − 0.269i)14-s + (−0.877 − 0.480i)15-s + (0.228 − 0.973i)16-s + (0.126 + 0.991i)17-s + ⋯

Functional equation

Λ(s)=(967s/2 ΓR(s) L(s)=((−0.492+0.870i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(967s/2 ΓR(s) L(s)=((−0.492+0.870i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 967967
Sign: −0.492+0.870i-0.492 + 0.870i
Analytic conductor: 4.490724.49072
Root analytic conductor: 4.490724.49072
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ967(16,⋅)\chi_{967} (16, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 967, (0: ), −0.492+0.870i)(1,\ 967,\ (0:\ ),\ -0.492 + 0.870i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.9129324062+1.564617299i0.9129324062 + 1.564617299i
L(12)L(\frac12) ≈\approx 0.9129324062+1.564617299i0.9129324062 + 1.564617299i
L(1)L(1) ≈\approx 1.115675897+0.7717905963i1.115675897 + 0.7717905963i
L(1)L(1) ≈\approx 1.115675897+0.7717905963i1.115675897 + 0.7717905963i

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+(0.328+0.944i)T 1 + (0.328 + 0.944i)T
3 1+(0.981+0.193i)T 1 + (0.981 + 0.193i)T
5 1+(−0.953−0.300i)T 1 + (-0.953 - 0.300i)T
7 1+(0.0617−0.998i)T 1 + (0.0617 - 0.998i)T
11 1+(0.279+0.960i)T 1 + (0.279 + 0.960i)T
13 1+(−0.971+0.238i)T 1 + (-0.971 + 0.238i)T
17 1+(0.126+0.991i)T 1 + (0.126 + 0.991i)T
19 1+(0.919−0.392i)T 1 + (0.919 - 0.392i)T
23 1+(0.996+0.0779i)T 1 + (0.996 + 0.0779i)T
29 1+(−0.998−0.0585i)T 1 + (-0.998 - 0.0585i)T
31 1+(0.754+0.655i)T 1 + (0.754 + 0.655i)T
37 1+(0.771−0.636i)T 1 + (0.771 - 0.636i)T
41 1+(−0.775−0.631i)T 1 + (-0.775 - 0.631i)T
43 1+(−0.767+0.641i)T 1 + (-0.767 + 0.641i)T
47 1+(0.581+0.813i)T 1 + (0.581 + 0.813i)T
53 1+(0.291+0.956i)T 1 + (0.291 + 0.956i)T
59 1+(0.152+0.988i)T 1 + (0.152 + 0.988i)T
61 1+(−0.158+0.987i)T 1 + (-0.158 + 0.987i)T
67 1+(−0.945−0.325i)T 1 + (-0.945 - 0.325i)T
71 1+(0.653+0.756i)T 1 + (0.653 + 0.756i)T
73 1+(0.291−0.956i)T 1 + (0.291 - 0.956i)T
79 1+(−0.997−0.0714i)T 1 + (-0.997 - 0.0714i)T
83 1+(0.861+0.508i)T 1 + (0.861 + 0.508i)T
89 1+(0.190+0.981i)T 1 + (0.190 + 0.981i)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

−21.48016020563478919623246361305, −20.463521697613182987340533137798, −20.05150681782591532715478446917, −18.96100028478972403368955165971, −18.87339167522529158748469528266, −18.14387069320723869822126517354, −16.678739583271747972409003526317, −15.51707128635502648940038948338, −14.98486834436912773294427975134, −14.32365487554280612150268331573, −13.47245508691856281680594046644, −12.62216231801488044648538756814, −11.73752952178963292960663844260, −11.43447077651271588018761865706, −10.05880661975177601295966759022, −9.359285702371401247380396505638, −8.55201939983166503674189160000, −7.82851791586193001991694867672, −6.74984487967888255171459367902, −5.4280055829066998219921340356, −4.58695108706651375638662421510, −3.29937269076125163480114113423, −3.08604481593505682089014477331, −2.07234628646028376565132495055, −0.70596449832211097547702793898, 1.23247093966021993637066957424, 2.87845582599370740692766994135, 3.8392118606666755870383794934, 4.405061392128957674373426273660, 5.09841185372184019708309487263, 6.7955878555493381341485042372, 7.42766522325098216640302787830, 7.79080543579455885765613514032, 8.89061555678801293303580611412, 9.545718341908134970595415618175, 10.52829390492753122451304547260, 11.88410939871168373722940274563, 12.6995423117386030772301043347, 13.3813474983266309808965177540, 14.304995963849764905603611706582, 14.9813167384234454461940244952, 15.39203279086274243715210957249, 16.46530360484355152848849225895, 16.97426522612937378729630066442, 17.90380412374408318420205957886, 19.081878313440287891116592003562, 19.695009888754477781273663867240, 20.37882294978440188680423102471, 21.183537611297803736660875145217, 22.17491172184570627435466976627

Graph of the ZZ-function along the critical line