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Properties

Label 2-296-296.101-c1-0-2
Degree 22
Conductor 296296
Sign 0.908−0.418i0.908 - 0.418i
Analytic cond. 2.363572.36357
Root an. cond. 1.537391.53739
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  + (−1.00 − 0.994i)2-s + (−2.16 + 1.24i)3-s + (0.0206 + 1.99i)4-s + (−0.890 − 1.54i)5-s + (3.41 + 0.896i)6-s + (−1.76 − 3.05i)7-s + (1.96 − 2.03i)8-s + (1.61 − 2.79i)9-s + (−0.639 + 2.43i)10-s + 4.13i·11-s + (−2.54 − 4.29i)12-s + (−0.0436 − 0.0756i)13-s + (−1.26 + 4.81i)14-s + (3.85 + 2.22i)15-s + (−3.99 + 0.0824i)16-s + (6.70 + 3.87i)17-s + ⋯
L(s)  = 1  + (−0.710 − 0.703i)2-s + (−1.24 + 0.720i)3-s + (0.0103 + 0.999i)4-s + (−0.398 − 0.689i)5-s + (1.39 + 0.365i)6-s + (−0.665 − 1.15i)7-s + (0.696 − 0.717i)8-s + (0.538 − 0.932i)9-s + (−0.202 + 0.770i)10-s + 1.24i·11-s + (−0.733 − 1.24i)12-s + (−0.0121 − 0.0209i)13-s + (−0.338 + 1.28i)14-s + (0.994 + 0.574i)15-s + (−0.999 + 0.0206i)16-s + (1.62 + 0.939i)17-s + ⋯

Functional equation

Λ(s)=(296s/2 ΓC(s) L(s)=((0.908−0.418i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(296s/2 ΓC(s+1/2) L(s)=((0.908−0.418i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 296296    =    23⋅372^{3} \cdot 37
Sign: 0.908−0.418i0.908 - 0.418i
Analytic conductor: 2.363572.36357
Root analytic conductor: 1.537391.53739
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ296(101,⋅)\chi_{296} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 296, ( :1/2), 0.908−0.418i)(2,\ 296,\ (\ :1/2),\ 0.908 - 0.418i)

Particular Values

L(1)L(1) ≈\approx 0.438502+0.0962400i0.438502 + 0.0962400i
L(12)L(\frac12) ≈\approx 0.438502+0.0962400i0.438502 + 0.0962400i
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+(1.00+0.994i)T 1 + (1.00 + 0.994i)T
37 1+(−3.42+5.02i)T 1 + (-3.42 + 5.02i)T
good3 1+(2.16−1.24i)T+(1.5−2.59i)T2 1 + (2.16 - 1.24i)T + (1.5 - 2.59i)T^{2}
5 1+(0.890+1.54i)T+(−2.5+4.33i)T2 1 + (0.890 + 1.54i)T + (-2.5 + 4.33i)T^{2}
7 1+(1.76+3.05i)T+(−3.5+6.06i)T2 1 + (1.76 + 3.05i)T + (-3.5 + 6.06i)T^{2}
11 1−4.13iT−11T2 1 - 4.13iT - 11T^{2}
13 1+(0.0436+0.0756i)T+(−6.5+11.2i)T2 1 + (0.0436 + 0.0756i)T + (-6.5 + 11.2i)T^{2}
17 1+(−6.70−3.87i)T+(8.5+14.7i)T2 1 + (-6.70 - 3.87i)T + (8.5 + 14.7i)T^{2}
19 1+(−2.34−4.05i)T+(−9.5+16.4i)T2 1 + (-2.34 - 4.05i)T + (-9.5 + 16.4i)T^{2}
23 1−4.36iT−23T2 1 - 4.36iT - 23T^{2}
29 1+3.98T+29T2 1 + 3.98T + 29T^{2}
31 1−1.77iT−31T2 1 - 1.77iT - 31T^{2}
41 1+(−4.07−7.06i)T+(−20.5+35.5i)T2 1 + (-4.07 - 7.06i)T + (-20.5 + 35.5i)T^{2}
43 1−5.20T+43T2 1 - 5.20T + 43T^{2}
47 1+4.78T+47T2 1 + 4.78T + 47T^{2}
53 1+(−8.84−5.10i)T+(26.5+45.8i)T2 1 + (-8.84 - 5.10i)T + (26.5 + 45.8i)T^{2}
59 1+(−1.80+3.12i)T+(−29.5−51.0i)T2 1 + (-1.80 + 3.12i)T + (-29.5 - 51.0i)T^{2}
61 1+(−0.507−0.879i)T+(−30.5+52.8i)T2 1 + (-0.507 - 0.879i)T + (-30.5 + 52.8i)T^{2}
67 1+(−2.90+1.67i)T+(33.5−58.0i)T2 1 + (-2.90 + 1.67i)T + (33.5 - 58.0i)T^{2}
71 1+(−7.96−13.7i)T+(−35.5+61.4i)T2 1 + (-7.96 - 13.7i)T + (-35.5 + 61.4i)T^{2}
73 1+14.8T+73T2 1 + 14.8T + 73T^{2}
79 1+(−3.90+2.25i)T+(39.5−68.4i)T2 1 + (-3.90 + 2.25i)T + (39.5 - 68.4i)T^{2}
83 1+(7.49+4.32i)T+(41.5+71.8i)T2 1 + (7.49 + 4.32i)T + (41.5 + 71.8i)T^{2}
89 1+(7.29+4.21i)T+(44.5+77.0i)T2 1 + (7.29 + 4.21i)T + (44.5 + 77.0i)T^{2}
97 1−9.43iT−97T2 1 - 9.43iT - 97T^{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

−11.75513319593748603039745836968, −10.71403948182712390757755525852, −10.00286973998297247143019361562, −9.602681725557747198459262584136, −7.976355561499935584249827531065, −7.19643797466847152215149418948, −5.71417358908360539642526582064, −4.38328434160037812905414984202, −3.69158127480911962093431661275, −1.10285610436915826997111163068, 0.62525870046369957490939122784, 2.90739959048419881529054336403, 5.35958945115633516708427286219, 5.86753513295907513770551068790, 6.78758656392042722857818757675, 7.55992820108838619114371482691, 8.778863347856258080780461110472, 9.760937029227156605875762142438, 10.98067534102914517470817856774, 11.50120601402248711389732891146

Graph of the ZZ-function along the critical line