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Properties

Label 2-2624-328.13-c0-0-0
Degree $2$
Conductor $2624$
Sign $0.465 + 0.885i$
Analytic cond. $1.30954$
Root an. cond. $1.14435$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.40 − 0.581i)3-s + (0.928 + 0.928i)9-s + (−0.398 + 0.243i)11-s + (0.178 − 0.744i)17-s + (0.0123 + 0.156i)19-s + (0.951 + 0.309i)25-s + (−0.182 − 0.439i)27-s + (0.701 − 0.111i)33-s + (−0.156 + 0.987i)41-s + (0.863 − 1.69i)43-s + (−0.156 − 0.987i)49-s + (−0.684 + 0.941i)51-s + (0.0737 − 0.226i)57-s + (1.34 − 0.437i)59-s + (0.891 − 1.45i)67-s + ⋯
L(s)  = 1  + (−1.40 − 0.581i)3-s + (0.928 + 0.928i)9-s + (−0.398 + 0.243i)11-s + (0.178 − 0.744i)17-s + (0.0123 + 0.156i)19-s + (0.951 + 0.309i)25-s + (−0.182 − 0.439i)27-s + (0.701 − 0.111i)33-s + (−0.156 + 0.987i)41-s + (0.863 − 1.69i)43-s + (−0.156 − 0.987i)49-s + (−0.684 + 0.941i)51-s + (0.0737 − 0.226i)57-s + (1.34 − 0.437i)59-s + (0.891 − 1.45i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.465 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.465 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $0.465 + 0.885i$
Analytic conductor: \(1.30954\)
Root analytic conductor: \(1.14435\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2624} (2145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2624,\ (\ :0),\ 0.465 + 0.885i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6702934812\)
\(L(\frac12)\) \(\approx\) \(0.6702934812\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 + (0.156 - 0.987i)T \)
good3 \( 1 + (1.40 + 0.581i)T + (0.707 + 0.707i)T^{2} \)
5 \( 1 + (-0.951 - 0.309i)T^{2} \)
7 \( 1 + (0.156 + 0.987i)T^{2} \)
11 \( 1 + (0.398 - 0.243i)T + (0.453 - 0.891i)T^{2} \)
13 \( 1 + (-0.987 - 0.156i)T^{2} \)
17 \( 1 + (-0.178 + 0.744i)T + (-0.891 - 0.453i)T^{2} \)
19 \( 1 + (-0.0123 - 0.156i)T + (-0.987 + 0.156i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (0.891 - 0.453i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + (-0.863 + 1.69i)T + (-0.587 - 0.809i)T^{2} \)
47 \( 1 + (0.156 - 0.987i)T^{2} \)
53 \( 1 + (-0.891 + 0.453i)T^{2} \)
59 \( 1 + (-1.34 + 0.437i)T + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.587 - 0.809i)T^{2} \)
67 \( 1 + (-0.891 + 1.45i)T + (-0.453 - 0.891i)T^{2} \)
71 \( 1 + (0.453 - 0.891i)T^{2} \)
73 \( 1 + (0.437 - 0.437i)T - iT^{2} \)
79 \( 1 + (0.707 + 0.707i)T^{2} \)
83 \( 1 - 1.17iT - T^{2} \)
89 \( 1 + (1.29 + 1.10i)T + (0.156 + 0.987i)T^{2} \)
97 \( 1 + (-1.04 + 1.70i)T + (-0.453 - 0.891i)T^{2} \)
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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

−8.899208000725909011774784041501, −7.996666327190925871202049852779, −7.07955319861435375266241149420, −6.76966352341937550870860786757, −5.70776076128765700544120836998, −5.25382487540509091759157422378, −4.43734527030200934226740633933, −3.16291692228187993791732942353, −1.93427777478118049394413853545, −0.67722598940912859673041771093, 1.01247520895196650385612058042, 2.59917729403724179774519347547, 3.80682855217925749447386036918, 4.61019669510476177106440276721, 5.32781113510743498982470854510, 6.00490330329518501781357243526, 6.63935792049394988766116198253, 7.58721952070047669264303473391, 8.479697414987362925797395255793, 9.303844196809183854341945956666

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