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Properties

Label 2-116-116.19-c1-0-1
Degree 22
Conductor 116116
Sign 0.164−0.986i0.164 - 0.986i
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  + (0.810 + 1.15i)2-s + (−1.24 − 0.436i)3-s + (−0.687 + 1.87i)4-s + (1.45 + 0.332i)5-s + (−0.504 − 1.79i)6-s + (2.16 + 4.49i)7-s + (−2.73 + 0.724i)8-s + (−0.982 − 0.783i)9-s + (0.795 + 1.95i)10-s + (0.429 − 3.81i)11-s + (1.67 − 2.04i)12-s + (0.621 − 0.495i)13-s + (−3.45 + 6.14i)14-s + (−1.67 − 1.05i)15-s + (−3.05 − 2.58i)16-s + (2.22 − 2.22i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.819i)2-s + (−0.719 − 0.251i)3-s + (−0.343 + 0.939i)4-s + (0.651 + 0.148i)5-s + (−0.205 − 0.734i)6-s + (0.817 + 1.69i)7-s + (−0.966 + 0.256i)8-s + (−0.327 − 0.261i)9-s + (0.251 + 0.619i)10-s + (0.129 − 1.15i)11-s + (0.483 − 0.589i)12-s + (0.172 − 0.137i)13-s + (−0.923 + 1.64i)14-s + (−0.431 − 0.271i)15-s + (−0.763 − 0.645i)16-s + (0.539 − 0.539i)17-s + ⋯

Functional equation

Λ(s)=(116s/2 ΓC(s) L(s)=((0.164−0.986i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(116s/2 ΓC(s+1/2) L(s)=((0.164−0.986i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 116116    =    22⋅292^{2} \cdot 29
Sign: 0.164−0.986i0.164 - 0.986i
Analytic conductor: 0.9262640.926264
Root analytic conductor: 0.9624260.962426
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ116(19,⋅)\chi_{116} (19, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 116, ( :1/2), 0.164−0.986i)(2,\ 116,\ (\ :1/2),\ 0.164 - 0.986i)

Particular Values

L(1)L(1) ≈\approx 0.914415+0.774356i0.914415 + 0.774356i
L(12)L(\frac12) ≈\approx 0.914415+0.774356i0.914415 + 0.774356i
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+(−0.810−1.15i)T 1 + (-0.810 - 1.15i)T
29 1+(−3.73−3.87i)T 1 + (-3.73 - 3.87i)T
good3 1+(1.24+0.436i)T+(2.34+1.87i)T2 1 + (1.24 + 0.436i)T + (2.34 + 1.87i)T^{2}
5 1+(−1.45−0.332i)T+(4.50+2.16i)T2 1 + (-1.45 - 0.332i)T + (4.50 + 2.16i)T^{2}
7 1+(−2.16−4.49i)T+(−4.36+5.47i)T2 1 + (-2.16 - 4.49i)T + (-4.36 + 5.47i)T^{2}
11 1+(−0.429+3.81i)T+(−10.7−2.44i)T2 1 + (-0.429 + 3.81i)T + (-10.7 - 2.44i)T^{2}
13 1+(−0.621+0.495i)T+(2.89−12.6i)T2 1 + (-0.621 + 0.495i)T + (2.89 - 12.6i)T^{2}
17 1+(−2.22+2.22i)T−17iT2 1 + (-2.22 + 2.22i)T - 17iT^{2}
19 1+(1.65+4.73i)T+(−14.8+11.8i)T2 1 + (1.65 + 4.73i)T + (-14.8 + 11.8i)T^{2}
23 1+(−3.02+0.689i)T+(20.7−9.97i)T2 1 + (-3.02 + 0.689i)T + (20.7 - 9.97i)T^{2}
31 1+(4.84−3.04i)T+(13.4−27.9i)T2 1 + (4.84 - 3.04i)T + (13.4 - 27.9i)T^{2}
37 1+(2.57−0.290i)T+(36.0−8.23i)T2 1 + (2.57 - 0.290i)T + (36.0 - 8.23i)T^{2}
41 1+(−0.977−0.977i)T+41iT2 1 + (-0.977 - 0.977i)T + 41iT^{2}
43 1+(−5.24+8.34i)T+(−18.6−38.7i)T2 1 + (-5.24 + 8.34i)T + (-18.6 - 38.7i)T^{2}
47 1+(7.11+0.801i)T+(45.8+10.4i)T2 1 + (7.11 + 0.801i)T + (45.8 + 10.4i)T^{2}
53 1+(−1.32+5.81i)T+(−47.7−22.9i)T2 1 + (-1.32 + 5.81i)T + (-47.7 - 22.9i)T^{2}
59 1−0.973iT−59T2 1 - 0.973iT - 59T^{2}
61 1+(1.52−4.36i)T+(−47.6−38.0i)T2 1 + (1.52 - 4.36i)T + (-47.6 - 38.0i)T^{2}
67 1+(2.79−3.50i)T+(−14.9−65.3i)T2 1 + (2.79 - 3.50i)T + (-14.9 - 65.3i)T^{2}
71 1+(−4.75−5.96i)T+(−15.7+69.2i)T2 1 + (-4.75 - 5.96i)T + (-15.7 + 69.2i)T^{2}
73 1+(5.74−9.13i)T+(−31.6−65.7i)T2 1 + (5.74 - 9.13i)T + (-31.6 - 65.7i)T^{2}
79 1+(−4.41+0.497i)T+(77.0−17.5i)T2 1 + (-4.41 + 0.497i)T + (77.0 - 17.5i)T^{2}
83 1+(0.599−1.24i)T+(−51.7−64.8i)T2 1 + (0.599 - 1.24i)T + (-51.7 - 64.8i)T^{2}
89 1+(−1.59−2.53i)T+(−38.6+80.1i)T2 1 + (-1.59 - 2.53i)T + (-38.6 + 80.1i)T^{2}
97 1+(3.39+9.69i)T+(−75.8+60.4i)T2 1 + (3.39 + 9.69i)T + (-75.8 + 60.4i)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

−13.97297772045545590025370623552, −12.74350191368714435026036138441, −11.85350722434606567148303475778, −11.13153320722085222783511584514, −9.047046986615733118084237179417, −8.480867623463370389699326298420, −6.75052146100344785105412926697, −5.70067931003739202480444393357, −5.24060629088224894717474839414, −2.87295430756270247983179889728, 1.62157176214043520539554049360, 4.04185047338083802674417729041, 4.97805570307254183634715987378, 6.20955562243585502485748337381, 7.80414830259939474431833974401, 9.671070478945394504896585206504, 10.45797751784808903595069770342, 11.11060243369636311262826548348, 12.22493107703637621520152680522, 13.32259828281875619443590589484

Graph of the ZZ-function along the critical line