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Properties

Label 2-920-5.4-c1-0-13
Degree $2$
Conductor $920$
Sign $0.906 + 0.423i$
Analytic cond. $7.34623$
Root an. cond. $2.71039$
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  − 3.20i·3-s + (2.02 + 0.945i)5-s + 4.54i·7-s − 7.29·9-s + 4.59·11-s + 5.17i·13-s + (3.03 − 6.50i)15-s − 3.33i·17-s + 4.43·19-s + 14.5·21-s + i·23-s + (3.21 + 3.83i)25-s + 13.8i·27-s − 4.13·29-s + 7.82·31-s + ⋯
L(s)  = 1  − 1.85i·3-s + (0.906 + 0.423i)5-s + 1.71i·7-s − 2.43·9-s + 1.38·11-s + 1.43i·13-s + (0.783 − 1.67i)15-s − 0.809i·17-s + 1.01·19-s + 3.18·21-s + 0.208i·23-s + (0.642 + 0.766i)25-s + 2.65i·27-s − 0.766·29-s + 1.40·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.423i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 + 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.906 + 0.423i$
Analytic conductor: \(7.34623\)
Root analytic conductor: \(2.71039\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :1/2),\ 0.906 + 0.423i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85834 - 0.412418i\)
\(L(\frac12)\) \(\approx\) \(1.85834 - 0.412418i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-2.02 - 0.945i)T \)
23 \( 1 - iT \)
good3 \( 1 + 3.20iT - 3T^{2} \)
7 \( 1 - 4.54iT - 7T^{2} \)
11 \( 1 - 4.59T + 11T^{2} \)
13 \( 1 - 5.17iT - 13T^{2} \)
17 \( 1 + 3.33iT - 17T^{2} \)
19 \( 1 - 4.43T + 19T^{2} \)
29 \( 1 + 4.13T + 29T^{2} \)
31 \( 1 - 7.82T + 31T^{2} \)
37 \( 1 - 1.03iT - 37T^{2} \)
41 \( 1 + 5.75T + 41T^{2} \)
43 \( 1 - 1.67iT - 43T^{2} \)
47 \( 1 + 6.20iT - 47T^{2} \)
53 \( 1 + 7.35iT - 53T^{2} \)
59 \( 1 + 1.83T + 59T^{2} \)
61 \( 1 - 0.524T + 61T^{2} \)
67 \( 1 - 2.55iT - 67T^{2} \)
71 \( 1 + 6.58T + 71T^{2} \)
73 \( 1 + 2.41iT - 73T^{2} \)
79 \( 1 - 9.85T + 79T^{2} \)
83 \( 1 + 10.7iT - 83T^{2} \)
89 \( 1 + 5.13T + 89T^{2} \)
97 \( 1 - 3.36iT - 97T^{2} \)
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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

−9.612649106585037468949258051557, −9.122006746892561409126068424387, −8.420857974426162718550185244007, −7.17513959773576315862333386596, −6.59233494257185718266107434783, −6.03344715237402053452904866791, −5.16488299348283905185178825600, −3.10759963164592093069671028013, −2.15723246134015602230845822952, −1.48702979283524138756525661691, 1.03944065497005211650234093339, 3.15431973944927297379704415503, 3.94180325591282773301594420451, 4.66784225444538387062313236432, 5.58970445199339764066942808851, 6.46318253083133177434756133368, 7.81626647513721943179902900508, 8.767095878234418256370024856479, 9.602055656104511083878853776669, 10.11518647091628847450123712402

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