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Properties

Label 2-368-16.13-c1-0-1
Degree $2$
Conductor $368$
Sign $0.826 - 0.563i$
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  + (−1.33 − 0.473i)2-s + (−1.96 − 1.96i)3-s + (1.55 + 1.26i)4-s + (1.87 − 1.87i)5-s + (1.68 + 3.54i)6-s + 4.01i·7-s + (−1.47 − 2.41i)8-s + 4.69i·9-s + (−3.38 + 1.60i)10-s + (−3.48 + 3.48i)11-s + (−0.569 − 5.51i)12-s + (2.06 + 2.06i)13-s + (1.89 − 5.34i)14-s − 7.34·15-s + (0.816 + 3.91i)16-s − 7.21·17-s + ⋯
L(s)  = 1  + (−0.942 − 0.334i)2-s + (−1.13 − 1.13i)3-s + (0.775 + 0.630i)4-s + (0.837 − 0.837i)5-s + (0.688 + 1.44i)6-s + 1.51i·7-s + (−0.520 − 0.854i)8-s + 1.56i·9-s + (−1.06 + 0.508i)10-s + (−1.05 + 1.05i)11-s + (−0.164 − 1.59i)12-s + (0.571 + 0.571i)13-s + (0.507 − 1.42i)14-s − 1.89·15-s + (0.204 + 0.978i)16-s − 1.75·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $0.826 - 0.563i$
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: \((2,\ 368,\ (\ :1/2),\ 0.826 - 0.563i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.453109 + 0.139734i\)
\(L(\frac12)\) \(\approx\) \(0.453109 + 0.139734i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 + 0.473i)T \)
23 \( 1 - iT \)
good3 \( 1 + (1.96 + 1.96i)T + 3iT^{2} \)
5 \( 1 + (-1.87 + 1.87i)T - 5iT^{2} \)
7 \( 1 - 4.01iT - 7T^{2} \)
11 \( 1 + (3.48 - 3.48i)T - 11iT^{2} \)
13 \( 1 + (-2.06 - 2.06i)T + 13iT^{2} \)
17 \( 1 + 7.21T + 17T^{2} \)
19 \( 1 + (-3.95 - 3.95i)T + 19iT^{2} \)
29 \( 1 + (-3.29 - 3.29i)T + 29iT^{2} \)
31 \( 1 + 1.19T + 31T^{2} \)
37 \( 1 + (-2.04 + 2.04i)T - 37iT^{2} \)
41 \( 1 - 2.62iT - 41T^{2} \)
43 \( 1 + (-1.95 + 1.95i)T - 43iT^{2} \)
47 \( 1 + 1.95T + 47T^{2} \)
53 \( 1 + (-2.99 + 2.99i)T - 53iT^{2} \)
59 \( 1 + (1.80 - 1.80i)T - 59iT^{2} \)
61 \( 1 + (2.96 + 2.96i)T + 61iT^{2} \)
67 \( 1 + (11.4 + 11.4i)T + 67iT^{2} \)
71 \( 1 + 3.18iT - 71T^{2} \)
73 \( 1 - 8.54iT - 73T^{2} \)
79 \( 1 - 2.41T + 79T^{2} \)
83 \( 1 + (-3.25 - 3.25i)T + 83iT^{2} \)
89 \( 1 - 13.6iT - 89T^{2} \)
97 \( 1 + 7.35T + 97T^{2} \)
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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

−11.61203739092337160523491584123, −10.70646085419982234838889197594, −9.514121114157154470920619291596, −8.846379853560526575220001664147, −7.80811862191055794662118456128, −6.69678663263242478068142405582, −5.88806233903434184874647058828, −5.01824817932377385838011472865, −2.31146901309633471396579133657, −1.58703069145562006954915266826, 0.49277328115454137281095632210, 2.93121762675695391607157438876, 4.57453973211891606204893396692, 5.75282481536550898980310128699, 6.43484546195094179793400067684, 7.40246095739413580237999837775, 8.741063879261495901129685574500, 9.913064545167868311254763543113, 10.43262049641897379398318540189, 10.95273263435851248576885886996

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