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Properties

Label 2-116-116.11-c1-0-1
Degree 22
Conductor 116116
Sign 0.341−0.939i0.341 - 0.939i
Analytic cond. 0.9262640.926264
Root an. cond. 0.9624260.962426
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.490 − 1.32i)2-s + (−2.34 + 0.264i)3-s + (−1.51 + 1.30i)4-s + (−0.588 + 1.22i)5-s + (1.50 + 2.98i)6-s + (2.56 + 2.04i)7-s + (2.47 + 1.37i)8-s + (2.52 − 0.575i)9-s + (1.91 + 0.181i)10-s + (−5.07 + 3.19i)11-s + (3.22 − 3.45i)12-s + (−1.29 − 0.296i)13-s + (1.45 − 4.39i)14-s + (1.05 − 3.02i)15-s + (0.615 − 3.95i)16-s + (−3.79 + 3.79i)17-s + ⋯
L(s)  = 1  + (−0.346 − 0.937i)2-s + (−1.35 + 0.152i)3-s + (−0.759 + 0.650i)4-s + (−0.263 + 0.546i)5-s + (0.613 + 1.21i)6-s + (0.968 + 0.772i)7-s + (0.873 + 0.486i)8-s + (0.840 − 0.191i)9-s + (0.604 + 0.0573i)10-s + (−1.53 + 0.962i)11-s + (0.930 − 0.998i)12-s + (−0.359 − 0.0821i)13-s + (0.388 − 1.17i)14-s + (0.273 − 0.781i)15-s + (0.153 − 0.988i)16-s + (−0.919 + 0.919i)17-s + ⋯

Functional equation

Λ(s)=(116s/2 ΓC(s) L(s)=((0.341−0.939i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.341 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(116s/2 ΓC(s+1/2) L(s)=((0.341−0.939i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.341 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 116116    =    22⋅292^{2} \cdot 29
Sign: 0.341−0.939i0.341 - 0.939i
Analytic conductor: 0.9262640.926264
Root analytic conductor: 0.9624260.962426
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ116(11,⋅)\chi_{116} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 116, ( :1/2), 0.341−0.939i)(2,\ 116,\ (\ :1/2),\ 0.341 - 0.939i)

Particular Values

L(1)L(1) ≈\approx 0.312544+0.218930i0.312544 + 0.218930i
L(12)L(\frac12) ≈\approx 0.312544+0.218930i0.312544 + 0.218930i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.490+1.32i)T 1 + (0.490 + 1.32i)T
29 1+(−2.93+4.51i)T 1 + (-2.93 + 4.51i)T
good3 1+(2.34−0.264i)T+(2.92−0.667i)T2 1 + (2.34 - 0.264i)T + (2.92 - 0.667i)T^{2}
5 1+(0.588−1.22i)T+(−3.11−3.90i)T2 1 + (0.588 - 1.22i)T + (-3.11 - 3.90i)T^{2}
7 1+(−2.56−2.04i)T+(1.55+6.82i)T2 1 + (-2.56 - 2.04i)T + (1.55 + 6.82i)T^{2}
11 1+(5.07−3.19i)T+(4.77−9.91i)T2 1 + (5.07 - 3.19i)T + (4.77 - 9.91i)T^{2}
13 1+(1.29+0.296i)T+(11.7+5.64i)T2 1 + (1.29 + 0.296i)T + (11.7 + 5.64i)T^{2}
17 1+(3.79−3.79i)T−17iT2 1 + (3.79 - 3.79i)T - 17iT^{2}
19 1+(−0.151+1.34i)T+(−18.5−4.22i)T2 1 + (-0.151 + 1.34i)T + (-18.5 - 4.22i)T^{2}
23 1+(−1.29−2.68i)T+(−14.3+17.9i)T2 1 + (-1.29 - 2.68i)T + (-14.3 + 17.9i)T^{2}
31 1+(1.44+4.14i)T+(−24.2+19.3i)T2 1 + (1.44 + 4.14i)T + (-24.2 + 19.3i)T^{2}
37 1+(3.02−4.81i)T+(−16.0−33.3i)T2 1 + (3.02 - 4.81i)T + (-16.0 - 33.3i)T^{2}
41 1+(4.55+4.55i)T+41iT2 1 + (4.55 + 4.55i)T + 41iT^{2}
43 1+(−6.65−2.33i)T+(33.6+26.8i)T2 1 + (-6.65 - 2.33i)T + (33.6 + 26.8i)T^{2}
47 1+(0.260+0.414i)T+(−20.3+42.3i)T2 1 + (0.260 + 0.414i)T + (-20.3 + 42.3i)T^{2}
53 1+(−5.75−2.77i)T+(33.0+41.4i)T2 1 + (-5.75 - 2.77i)T + (33.0 + 41.4i)T^{2}
59 1+6.26iT−59T2 1 + 6.26iT - 59T^{2}
61 1+(−1.38−12.2i)T+(−59.4+13.5i)T2 1 + (-1.38 - 12.2i)T + (-59.4 + 13.5i)T^{2}
67 1+(−0.856−3.75i)T+(−60.3+29.0i)T2 1 + (-0.856 - 3.75i)T + (-60.3 + 29.0i)T^{2}
71 1+(2.73−11.9i)T+(−63.9−30.8i)T2 1 + (2.73 - 11.9i)T + (-63.9 - 30.8i)T^{2}
73 1+(−7.19−2.51i)T+(57.0+45.5i)T2 1 + (-7.19 - 2.51i)T + (57.0 + 45.5i)T^{2}
79 1+(−2.90+4.62i)T+(−34.2−71.1i)T2 1 + (-2.90 + 4.62i)T + (-34.2 - 71.1i)T^{2}
83 1+(2.50−1.99i)T+(18.4−80.9i)T2 1 + (2.50 - 1.99i)T + (18.4 - 80.9i)T^{2}
89 1+(−12.1+4.23i)T+(69.5−55.4i)T2 1 + (-12.1 + 4.23i)T + (69.5 - 55.4i)T^{2}
97 1+(−0.141+1.25i)T+(−94.5−21.5i)T2 1 + (-0.141 + 1.25i)T + (-94.5 - 21.5i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−13.27529317641222227306832421421, −12.33539607687439119492156043921, −11.48604338972315137737783879493, −10.84057274976576526471497506241, −10.04491617764103710375010495664, −8.476487547999900586495623286559, −7.29787523402278920438672632460, −5.46175720496265322153308624222, −4.58025810736369975644233136203, −2.35301355645904820615558320807, 0.54925817958332078324495285454, 4.77114294217893018285801994487, 5.24643588503547898886961615030, 6.68069057404575526009919823164, 7.76449064290503183991704218100, 8.754909729854625994031117013443, 10.56030603884577047683239302627, 10.93731770910138042664429062204, 12.33510696412655165691431732871, 13.45005779427224207853718031276

Graph of the ZZ-function along the critical line