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Properties

Label 2-368-16.5-c1-0-29
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: $\chi_{368} (277, \cdot )$
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

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

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