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Properties

Label 2-368-16.5-c1-0-28
Degree 22
Conductor 368368
Sign −0.991+0.130i-0.991 + 0.130i
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 − 0.366i)2-s + (−0.633 + 0.633i)3-s + (1.73 + i)4-s + (−1 − i)5-s + (1.09 − 0.633i)6-s + 2.73i·7-s + (−1.99 − 2i)8-s + 2.19i·9-s + (1 + 1.73i)10-s + (−4.46 − 4.46i)11-s + (−1.73 + 0.464i)12-s + (2.09 − 2.09i)13-s + (1 − 3.73i)14-s + 1.26·15-s + (1.99 + 3.46i)16-s − 1.26·17-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (−0.366 + 0.366i)3-s + (0.866 + 0.5i)4-s + (−0.447 − 0.447i)5-s + (0.448 − 0.258i)6-s + 1.03i·7-s + (−0.707 − 0.707i)8-s + 0.732i·9-s + (0.316 + 0.547i)10-s + (−1.34 − 1.34i)11-s + (−0.499 + 0.133i)12-s + (0.581 − 0.581i)13-s + (0.267 − 0.997i)14-s + 0.327·15-s + (0.499 + 0.866i)16-s − 0.307·17-s + ⋯

Functional equation

Λ(s)=(368s/2 ΓC(s) L(s)=((−0.991+0.130i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(368s/2 ΓC(s+1/2) L(s)=((−0.991+0.130i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 368368    =    24⋅232^{4} \cdot 23
Sign: −0.991+0.130i-0.991 + 0.130i
Analytic conductor: 2.938492.93849
Root analytic conductor: 1.714201.71420
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ368(277,⋅)\chi_{368} (277, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 368, ( :1/2), −0.991+0.130i)(2,\ 368,\ (\ :1/2),\ -0.991 + 0.130i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+(1.36+0.366i)T 1 + (1.36 + 0.366i)T
23 1+iT 1 + iT
good3 1+(0.633−0.633i)T−3iT2 1 + (0.633 - 0.633i)T - 3iT^{2}
5 1+(1+i)T+5iT2 1 + (1 + i)T + 5iT^{2}
7 1−2.73iT−7T2 1 - 2.73iT - 7T^{2}
11 1+(4.46+4.46i)T+11iT2 1 + (4.46 + 4.46i)T + 11iT^{2}
13 1+(−2.09+2.09i)T−13iT2 1 + (-2.09 + 2.09i)T - 13iT^{2}
17 1+1.26T+17T2 1 + 1.26T + 17T^{2}
19 1+(2.73−2.73i)T−19iT2 1 + (2.73 - 2.73i)T - 19iT^{2}
29 1+(3.09−3.09i)T−29iT2 1 + (3.09 - 3.09i)T - 29iT^{2}
31 1+10.4T+31T2 1 + 10.4T + 31T^{2}
37 1+(7.73+7.73i)T+37iT2 1 + (7.73 + 7.73i)T + 37iT^{2}
41 1−3iT−41T2 1 - 3iT - 41T^{2}
43 1+(−1.46−1.46i)T+43iT2 1 + (-1.46 - 1.46i)T + 43iT^{2}
47 1+11.7T+47T2 1 + 11.7T + 47T^{2}
53 1+(−0.196−0.196i)T+53iT2 1 + (-0.196 - 0.196i)T + 53iT^{2}
59 1+(−10.4−10.4i)T+59iT2 1 + (-10.4 - 10.4i)T + 59iT^{2}
61 1+(−2+2i)T−61iT2 1 + (-2 + 2i)T - 61iT^{2}
67 1+(−0.464+0.464i)T−67iT2 1 + (-0.464 + 0.464i)T - 67iT^{2}
71 1−12.4iT−71T2 1 - 12.4iT - 71T^{2}
73 1+7iT−73T2 1 + 7iT - 73T^{2}
79 1−2.92T+79T2 1 - 2.92T + 79T^{2}
83 1+(−6.66+6.66i)T−83iT2 1 + (-6.66 + 6.66i)T - 83iT^{2}
89 1+10.3iT−89T2 1 + 10.3iT - 89T^{2}
97 1−12.7T+97T2 1 - 12.7T + 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

−10.85307897105858936636323630191, −10.30177987036213550623997860827, −8.849662191488841616490449213421, −8.424292298328366660082524853685, −7.61348576143554426044131285646, −5.96464113274735076046139986862, −5.30200746853965537319985886805, −3.55087930145378813392451357044, −2.23242126919152895191462677501, 0, 1.88483135490906404425815753519, 3.65429588468744074944968733620, 5.19849533992952265533046248259, 6.68716037223693216751768882518, 7.09042418687421295851126375208, 7.890758913483899289665927792645, 9.136755604008863120337862852930, 10.05456586925779227816698797200, 10.89277660402553851334524050135

Graph of the ZZ-function along the critical line