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Properties

Label 2-368-16.13-c1-0-10
Degree 22
Conductor 368368
Sign 0.515−0.856i0.515 - 0.856i
Analytic cond. 2.938492.93849
Root an. cond. 1.714201.71420
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  + (−0.741 − 1.20i)2-s + (1.96 + 1.96i)3-s + (−0.900 + 1.78i)4-s + (−0.463 + 0.463i)5-s + (0.908 − 3.81i)6-s + 1.33i·7-s + (2.81 − 0.238i)8-s + 4.69i·9-s + (0.900 + 0.214i)10-s + (−1.06 + 1.06i)11-s + (−5.26 + 1.73i)12-s + (−0.743 − 0.743i)13-s + (1.61 − 0.992i)14-s − 1.81·15-s + (−2.37 − 3.21i)16-s − 1.89·17-s + ⋯
L(s)  = 1  + (−0.524 − 0.851i)2-s + (1.13 + 1.13i)3-s + (−0.450 + 0.892i)4-s + (−0.207 + 0.207i)5-s + (0.370 − 1.55i)6-s + 0.506i·7-s + (0.996 − 0.0843i)8-s + 1.56i·9-s + (0.284 + 0.0678i)10-s + (−0.319 + 0.319i)11-s + (−1.52 + 0.500i)12-s + (−0.206 − 0.206i)13-s + (0.431 − 0.265i)14-s − 0.468·15-s + (−0.594 − 0.804i)16-s − 0.460·17-s + ⋯

Functional equation

Λ(s)=(368s/2 ΓC(s) L(s)=((0.515−0.856i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 - 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(368s/2 ΓC(s+1/2) L(s)=((0.515−0.856i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.515 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 368368    =    24⋅232^{4} \cdot 23
Sign: 0.515−0.856i0.515 - 0.856i
Analytic conductor: 2.938492.93849
Root analytic conductor: 1.714201.71420
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ368(93,⋅)\chi_{368} (93, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 368, ( :1/2), 0.515−0.856i)(2,\ 368,\ (\ :1/2),\ 0.515 - 0.856i)

Particular Values

L(1)L(1) ≈\approx 1.13585+0.641968i1.13585 + 0.641968i
L(12)L(\frac12) ≈\approx 1.13585+0.641968i1.13585 + 0.641968i
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+(0.741+1.20i)T 1 + (0.741 + 1.20i)T
23 1+iT 1 + iT
good3 1+(−1.96−1.96i)T+3iT2 1 + (-1.96 - 1.96i)T + 3iT^{2}
5 1+(0.463−0.463i)T−5iT2 1 + (0.463 - 0.463i)T - 5iT^{2}
7 1−1.33iT−7T2 1 - 1.33iT - 7T^{2}
11 1+(1.06−1.06i)T−11iT2 1 + (1.06 - 1.06i)T - 11iT^{2}
13 1+(0.743+0.743i)T+13iT2 1 + (0.743 + 0.743i)T + 13iT^{2}
17 1+1.89T+17T2 1 + 1.89T + 17T^{2}
19 1+(−4.25−4.25i)T+19iT2 1 + (-4.25 - 4.25i)T + 19iT^{2}
29 1+(2.48+2.48i)T+29iT2 1 + (2.48 + 2.48i)T + 29iT^{2}
31 1−1.53T+31T2 1 - 1.53T + 31T^{2}
37 1+(0.463−0.463i)T−37iT2 1 + (0.463 - 0.463i)T - 37iT^{2}
41 1−1.58iT−41T2 1 - 1.58iT - 41T^{2}
43 1+(−2.95+2.95i)T−43iT2 1 + (-2.95 + 2.95i)T - 43iT^{2}
47 1−6.03T+47T2 1 - 6.03T + 47T^{2}
53 1+(−7.20+7.20i)T−53iT2 1 + (-7.20 + 7.20i)T - 53iT^{2}
59 1+(−9.14+9.14i)T−59iT2 1 + (-9.14 + 9.14i)T - 59iT^{2}
61 1+(2.74+2.74i)T+61iT2 1 + (2.74 + 2.74i)T + 61iT^{2}
67 1+(0.407+0.407i)T+67iT2 1 + (0.407 + 0.407i)T + 67iT^{2}
71 1+15.9iT−71T2 1 + 15.9iT - 71T^{2}
73 1−6.53iT−73T2 1 - 6.53iT - 73T^{2}
79 1+9.08T+79T2 1 + 9.08T + 79T^{2}
83 1+(11.9+11.9i)T+83iT2 1 + (11.9 + 11.9i)T + 83iT^{2}
89 1−11.9iT−89T2 1 - 11.9iT - 89T^{2}
97 1+8.92T+97T2 1 + 8.92T + 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

−11.33853755043458139779288064268, −10.36417996121122371142390291609, −9.745528094495017791755197166108, −9.015134756316643859837106046614, −8.201972016641441404492656845976, −7.36315320839416709862940117806, −5.29856743403655241437701460150, −4.10684804262063368598302028222, −3.24128829899481874042825773936, −2.22013890400501111066808199842, 0.989376765244284816075103307449, 2.59007737404211777848627304280, 4.26612886011434553147276122668, 5.72401332789039477256264373987, 7.12902295211146191500378819885, 7.29443617810433892491884802556, 8.437961348230923651974086311381, 8.939670132642160019238627108249, 10.00090594657659349151496040083, 11.17878466305453905424833225076

Graph of the ZZ-function along the critical line