Properties

Label 2-920-5.4-c1-0-14
Degree $2$
Conductor $920$
Sign $0.935 - 0.352i$
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  + 1.58i·3-s + (−2.09 + 0.787i)5-s − 2.84i·7-s + 0.493·9-s + 1.98·11-s − 4.69i·13-s + (−1.24 − 3.31i)15-s + 3.16i·17-s + 6.16·19-s + 4.50·21-s i·23-s + (3.75 − 3.29i)25-s + 5.53i·27-s + 6.61·29-s − 8.29·31-s + ⋯
L(s)  = 1  + 0.914i·3-s + (−0.935 + 0.352i)5-s − 1.07i·7-s + 0.164·9-s + 0.598·11-s − 1.30i·13-s + (−0.321 − 0.855i)15-s + 0.768i·17-s + 1.41·19-s + 0.983·21-s − 0.208i·23-s + (0.751 − 0.659i)25-s + 1.06i·27-s + 1.22·29-s − 1.49·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.935 - 0.352i$
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: \((2,\ 920,\ (\ :1/2),\ 0.935 - 0.352i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42666 + 0.259573i\)
\(L(\frac12)\) \(\approx\) \(1.42666 + 0.259573i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.09 - 0.787i)T \)
23 \( 1 + iT \)
good3 \( 1 - 1.58iT - 3T^{2} \)
7 \( 1 + 2.84iT - 7T^{2} \)
11 \( 1 - 1.98T + 11T^{2} \)
13 \( 1 + 4.69iT - 13T^{2} \)
17 \( 1 - 3.16iT - 17T^{2} \)
19 \( 1 - 6.16T + 19T^{2} \)
29 \( 1 - 6.61T + 29T^{2} \)
31 \( 1 + 8.29T + 31T^{2} \)
37 \( 1 + 1.71iT - 37T^{2} \)
41 \( 1 - 6.72T + 41T^{2} \)
43 \( 1 - 0.177iT - 43T^{2} \)
47 \( 1 - 11.4iT - 47T^{2} \)
53 \( 1 + 6.18iT - 53T^{2} \)
59 \( 1 - 7.61T + 59T^{2} \)
61 \( 1 - 11.8T + 61T^{2} \)
67 \( 1 - 3.07iT - 67T^{2} \)
71 \( 1 + 1.96T + 71T^{2} \)
73 \( 1 + 4.94iT - 73T^{2} \)
79 \( 1 + 8.53T + 79T^{2} \)
83 \( 1 + 3.49iT - 83T^{2} \)
89 \( 1 + 7.07T + 89T^{2} \)
97 \( 1 + 7.00iT - 97T^{2} \)
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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

−10.27069144141292646923100880800, −9.505757503565795834296850996530, −8.396201465818514232338389856609, −7.54853343654295636218222502624, −6.96204873753334986858541602404, −5.64789785064034266795479185859, −4.53575226444549353048139604048, −3.83256189385994783000852600614, −3.15621198229676692687130117766, −0.942123567764900871550747742166, 1.08820745558345285701988489394, 2.32738246876213597503285161610, 3.66898631213386309527397595504, 4.73455067530039023568847401227, 5.74214391672062257282862318281, 6.96183588175510068362863998971, 7.26607220586101610342826468654, 8.386620613051651416734956582018, 9.052395804338948081072777799072, 9.779970358277527884084430673241

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