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Properties

Label 2-116-116.11-c1-0-2
Degree 22
Conductor 116116
Sign −0.259−0.965i-0.259 - 0.965i
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  + (1.22 + 0.701i)2-s + (−2.69 + 0.304i)3-s + (1.01 + 1.72i)4-s + (−1.20 + 2.49i)5-s + (−3.52 − 1.51i)6-s + (0.625 + 0.498i)7-s + (0.0419 + 2.82i)8-s + (4.27 − 0.974i)9-s + (−3.22 + 2.22i)10-s + (4.00 − 2.51i)11-s + (−3.26 − 4.33i)12-s + (−2.59 − 0.593i)13-s + (0.418 + 1.05i)14-s + (2.48 − 7.09i)15-s + (−1.93 + 3.50i)16-s + (4.12 − 4.12i)17-s + ⋯
L(s)  = 1  + (0.868 + 0.495i)2-s + (−1.55 + 0.175i)3-s + (0.508 + 0.861i)4-s + (−0.537 + 1.11i)5-s + (−1.44 − 0.620i)6-s + (0.236 + 0.188i)7-s + (0.0148 + 0.999i)8-s + (1.42 − 0.324i)9-s + (−1.01 + 0.702i)10-s + (1.20 − 0.757i)11-s + (−0.943 − 1.25i)12-s + (−0.720 − 0.164i)13-s + (0.111 + 0.281i)14-s + (0.641 − 1.83i)15-s + (−0.482 + 0.875i)16-s + (0.999 − 0.999i)17-s + ⋯

Functional equation

Λ(s)=(116s/2 ΓC(s) L(s)=((−0.259−0.965i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(116s/2 ΓC(s+1/2) L(s)=((−0.259−0.965i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 116116    =    22⋅292^{2} \cdot 29
Sign: −0.259−0.965i-0.259 - 0.965i
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), −0.259−0.965i)(2,\ 116,\ (\ :1/2),\ -0.259 - 0.965i)

Particular Values

L(1)L(1) ≈\approx 0.622168+0.811376i0.622168 + 0.811376i
L(12)L(\frac12) ≈\approx 0.622168+0.811376i0.622168 + 0.811376i
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+(−1.22−0.701i)T 1 + (-1.22 - 0.701i)T
29 1+(−5.26+1.11i)T 1 + (-5.26 + 1.11i)T
good3 1+(2.69−0.304i)T+(2.92−0.667i)T2 1 + (2.69 - 0.304i)T + (2.92 - 0.667i)T^{2}
5 1+(1.20−2.49i)T+(−3.11−3.90i)T2 1 + (1.20 - 2.49i)T + (-3.11 - 3.90i)T^{2}
7 1+(−0.625−0.498i)T+(1.55+6.82i)T2 1 + (-0.625 - 0.498i)T + (1.55 + 6.82i)T^{2}
11 1+(−4.00+2.51i)T+(4.77−9.91i)T2 1 + (-4.00 + 2.51i)T + (4.77 - 9.91i)T^{2}
13 1+(2.59+0.593i)T+(11.7+5.64i)T2 1 + (2.59 + 0.593i)T + (11.7 + 5.64i)T^{2}
17 1+(−4.12+4.12i)T−17iT2 1 + (-4.12 + 4.12i)T - 17iT^{2}
19 1+(0.647−5.74i)T+(−18.5−4.22i)T2 1 + (0.647 - 5.74i)T + (-18.5 - 4.22i)T^{2}
23 1+(1.26+2.62i)T+(−14.3+17.9i)T2 1 + (1.26 + 2.62i)T + (-14.3 + 17.9i)T^{2}
31 1+(0.0981+0.280i)T+(−24.2+19.3i)T2 1 + (0.0981 + 0.280i)T + (-24.2 + 19.3i)T^{2}
37 1+(1.42−2.27i)T+(−16.0−33.3i)T2 1 + (1.42 - 2.27i)T + (-16.0 - 33.3i)T^{2}
41 1+(−6.66−6.66i)T+41iT2 1 + (-6.66 - 6.66i)T + 41iT^{2}
43 1+(1.02+0.357i)T+(33.6+26.8i)T2 1 + (1.02 + 0.357i)T + (33.6 + 26.8i)T^{2}
47 1+(1.28+2.03i)T+(−20.3+42.3i)T2 1 + (1.28 + 2.03i)T + (-20.3 + 42.3i)T^{2}
53 1+(12.1+5.87i)T+(33.0+41.4i)T2 1 + (12.1 + 5.87i)T + (33.0 + 41.4i)T^{2}
59 1+11.1iT−59T2 1 + 11.1iT - 59T^{2}
61 1+(0.603+5.35i)T+(−59.4+13.5i)T2 1 + (0.603 + 5.35i)T + (-59.4 + 13.5i)T^{2}
67 1+(−1.16−5.10i)T+(−60.3+29.0i)T2 1 + (-1.16 - 5.10i)T + (-60.3 + 29.0i)T^{2}
71 1+(1.67−7.31i)T+(−63.9−30.8i)T2 1 + (1.67 - 7.31i)T + (-63.9 - 30.8i)T^{2}
73 1+(3.02+1.05i)T+(57.0+45.5i)T2 1 + (3.02 + 1.05i)T + (57.0 + 45.5i)T^{2}
79 1+(−4.66+7.42i)T+(−34.2−71.1i)T2 1 + (-4.66 + 7.42i)T + (-34.2 - 71.1i)T^{2}
83 1+(−7.68+6.13i)T+(18.4−80.9i)T2 1 + (-7.68 + 6.13i)T + (18.4 - 80.9i)T^{2}
89 1+(1.83−0.641i)T+(69.5−55.4i)T2 1 + (1.83 - 0.641i)T + (69.5 - 55.4i)T^{2}
97 1+(0.198−1.75i)T+(−94.5−21.5i)T2 1 + (0.198 - 1.75i)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

−14.26041297001171127136728977030, −12.46296265935835899981622368367, −11.74104832130750908986644336405, −11.26359994286153971736812783225, −10.06899592400637404240729839942, −7.962531649249400781769343891339, −6.75706759287984815682178759637, −6.05048796987167520448717449927, −4.83047172703048842229430496239, −3.39837251111545633725365410661, 1.20342902098964184588748930687, 4.26530315459281268763933166286, 4.95017994113268478520913394718, 6.16602325942598006306019743287, 7.33026410619435051931590745540, 9.307213741162741938957608217327, 10.56223462017511877459190738078, 11.56388273480638456033825334279, 12.35396895111538154447320189370, 12.56308170508206744572573179322

Graph of the ZZ-function along the critical line