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Properties

Label 2-368-16.13-c1-0-34
Degree $2$
Conductor $368$
Sign $0.936 + 0.350i$
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  + (1.38 − 0.287i)2-s + (0.339 + 0.339i)3-s + (1.83 − 0.796i)4-s + (1.46 − 1.46i)5-s + (0.568 + 0.372i)6-s + 2.63i·7-s + (2.31 − 1.63i)8-s − 2.76i·9-s + (1.61 − 2.45i)10-s + (−2.99 + 2.99i)11-s + (0.893 + 0.352i)12-s + (−0.749 − 0.749i)13-s + (0.757 + 3.64i)14-s + 0.998·15-s + (2.72 − 2.92i)16-s − 3.64·17-s + ⋯
L(s)  = 1  + (0.979 − 0.203i)2-s + (0.196 + 0.196i)3-s + (0.917 − 0.398i)4-s + (0.657 − 0.657i)5-s + (0.231 + 0.152i)6-s + 0.994i·7-s + (0.816 − 0.576i)8-s − 0.923i·9-s + (0.509 − 0.777i)10-s + (−0.901 + 0.901i)11-s + (0.258 + 0.101i)12-s + (−0.207 − 0.207i)13-s + (0.202 + 0.973i)14-s + 0.257·15-s + (0.682 − 0.730i)16-s − 0.885·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $0.936 + 0.350i$
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: \((2,\ 368,\ (\ :1/2),\ 0.936 + 0.350i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.63665 - 0.477676i\)
\(L(\frac12)\) \(\approx\) \(2.63665 - 0.477676i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 0.287i)T \)
23 \( 1 + iT \)
good3 \( 1 + (-0.339 - 0.339i)T + 3iT^{2} \)
5 \( 1 + (-1.46 + 1.46i)T - 5iT^{2} \)
7 \( 1 - 2.63iT - 7T^{2} \)
11 \( 1 + (2.99 - 2.99i)T - 11iT^{2} \)
13 \( 1 + (0.749 + 0.749i)T + 13iT^{2} \)
17 \( 1 + 3.64T + 17T^{2} \)
19 \( 1 + (-1.76 - 1.76i)T + 19iT^{2} \)
29 \( 1 + (-0.0790 - 0.0790i)T + 29iT^{2} \)
31 \( 1 + 2.07T + 31T^{2} \)
37 \( 1 + (3.64 - 3.64i)T - 37iT^{2} \)
41 \( 1 - 1.90iT - 41T^{2} \)
43 \( 1 + (-6.84 + 6.84i)T - 43iT^{2} \)
47 \( 1 + 8.55T + 47T^{2} \)
53 \( 1 + (6.62 - 6.62i)T - 53iT^{2} \)
59 \( 1 + (5.45 - 5.45i)T - 59iT^{2} \)
61 \( 1 + (-2.13 - 2.13i)T + 61iT^{2} \)
67 \( 1 + (-3.51 - 3.51i)T + 67iT^{2} \)
71 \( 1 + 1.61iT - 71T^{2} \)
73 \( 1 + 14.4iT - 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 + (-1.26 - 1.26i)T + 83iT^{2} \)
89 \( 1 + 12.3iT - 89T^{2} \)
97 \( 1 - 7.54T + 97T^{2} \)
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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

−11.67331046134537991926051670959, −10.43055051161851484355695852987, −9.596587694967829241646983811956, −8.790614164973288290668028109915, −7.40141690428137971701284600117, −6.19792187276212811289909293263, −5.37176151774524990851548993173, −4.51571447069317620394101415623, −3.02976387776582137422094243857, −1.89314453486748937723889002260, 2.14647611322891899249383761149, 3.17806668114539975544731516127, 4.57073523551011673659973577738, 5.58455115885075479121458773326, 6.67541779815116205496210586145, 7.43952672869535936440227599559, 8.326986679551943293536942114158, 9.934659212313522878571839524724, 10.96201940963414219599887160547, 11.10960710558376953664746241946

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