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Properties

Label 2-368-16.13-c1-0-18
Degree $2$
Conductor $368$
Sign $0.510 - 0.860i$
Analytic cond. $2.93849$
Root an. cond. $1.71420$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.20 + 0.743i)2-s + (2.30 + 2.30i)3-s + (0.895 − 1.78i)4-s + (1.94 − 1.94i)5-s + (−4.48 − 1.05i)6-s − 3.32i·7-s + (0.252 + 2.81i)8-s + 7.60i·9-s + (−0.895 + 3.78i)10-s + (−1.35 + 1.35i)11-s + (6.17 − 2.05i)12-s + (−1.10 − 1.10i)13-s + (2.47 + 4.00i)14-s + 8.96·15-s + (−2.39 − 3.20i)16-s + 4.57·17-s + ⋯
L(s)  = 1  + (−0.850 + 0.525i)2-s + (1.32 + 1.32i)3-s + (0.447 − 0.894i)4-s + (0.870 − 0.870i)5-s + (−1.82 − 0.432i)6-s − 1.25i·7-s + (0.0891 + 0.996i)8-s + 2.53i·9-s + (−0.283 + 1.19i)10-s + (−0.408 + 0.408i)11-s + (1.78 − 0.593i)12-s + (−0.305 − 0.305i)13-s + (0.661 + 1.07i)14-s + 2.31·15-s + (−0.599 − 0.800i)16-s + 1.10·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 - 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.510 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $0.510 - 0.860i$
Analytic conductor: \(2.93849\)
Root analytic conductor: \(1.71420\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 368,\ (\ :1/2),\ 0.510 - 0.860i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36667 + 0.778281i\)
\(L(\frac12)\) \(\approx\) \(1.36667 + 0.778281i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.20 - 0.743i)T \)
23 \( 1 + iT \)
good3 \( 1 + (-2.30 - 2.30i)T + 3iT^{2} \)
5 \( 1 + (-1.94 + 1.94i)T - 5iT^{2} \)
7 \( 1 + 3.32iT - 7T^{2} \)
11 \( 1 + (1.35 - 1.35i)T - 11iT^{2} \)
13 \( 1 + (1.10 + 1.10i)T + 13iT^{2} \)
17 \( 1 - 4.57T + 17T^{2} \)
19 \( 1 + (-2.70 - 2.70i)T + 19iT^{2} \)
29 \( 1 + (-4.52 - 4.52i)T + 29iT^{2} \)
31 \( 1 + 9.00T + 31T^{2} \)
37 \( 1 + (0.166 - 0.166i)T - 37iT^{2} \)
41 \( 1 + 11.1iT - 41T^{2} \)
43 \( 1 + (1.75 - 1.75i)T - 43iT^{2} \)
47 \( 1 + 9.50T + 47T^{2} \)
53 \( 1 + (3.17 - 3.17i)T - 53iT^{2} \)
59 \( 1 + (-1.45 + 1.45i)T - 59iT^{2} \)
61 \( 1 + (9.07 + 9.07i)T + 61iT^{2} \)
67 \( 1 + (2.68 + 2.68i)T + 67iT^{2} \)
71 \( 1 - 3.42iT - 71T^{2} \)
73 \( 1 + 7.15iT - 73T^{2} \)
79 \( 1 + 6.02T + 79T^{2} \)
83 \( 1 + (-3.30 - 3.30i)T + 83iT^{2} \)
89 \( 1 + 8.26iT - 89T^{2} \)
97 \( 1 + 16.0T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.79570586661599448805649874546, −10.18568967527695655460149711211, −9.685955407573264955095331163145, −8.963558377448202208681362607037, −8.000552126367223269187124996785, −7.33745289084286792091530371004, −5.47158322461294417567298966531, −4.76883718055938436399819451647, −3.34500214807023177412935004351, −1.69515607631297657704701872804, 1.64893335918980742648644627753, 2.66184673809490647921617798013, 3.12801370766671834143326348038, 5.90043033186364399351628879007, 6.82463652400945031220146636004, 7.71505200661986847930275078336, 8.493232066887277978742159775380, 9.368220142526051822793270659520, 9.936627417494881450783400853813, 11.40271282980868454962753419627

Graph of the $Z$-function along the critical line