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Properties

Label 2-2624-328.229-c0-0-1
Degree $2$
Conductor $2624$
Sign $0.927 - 0.374i$
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  + (0.399 + 0.965i)3-s + (−0.0650 + 0.0650i)9-s + (−0.101 − 1.29i)11-s + (1.20 − 1.40i)17-s + (0.108 + 0.453i)19-s + (0.587 + 0.809i)25-s + (0.876 + 0.363i)27-s + (1.20 − 0.616i)33-s + (0.453 − 0.891i)41-s + (−1.16 + 0.183i)43-s + (0.453 + 0.891i)49-s + (1.83 + 0.596i)51-s + (−0.394 + 0.286i)57-s + (−0.831 + 1.14i)59-s + (0.156 + 0.0123i)67-s + ⋯
L(s)  = 1  + (0.399 + 0.965i)3-s + (−0.0650 + 0.0650i)9-s + (−0.101 − 1.29i)11-s + (1.20 − 1.40i)17-s + (0.108 + 0.453i)19-s + (0.587 + 0.809i)25-s + (0.876 + 0.363i)27-s + (1.20 − 0.616i)33-s + (0.453 − 0.891i)41-s + (−1.16 + 0.183i)43-s + (0.453 + 0.891i)49-s + (1.83 + 0.596i)51-s + (−0.394 + 0.286i)57-s + (−0.831 + 1.14i)59-s + (0.156 + 0.0123i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $0.927 - 0.374i$
Analytic conductor: \(1.30954\)
Root analytic conductor: \(1.14435\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2624} (1377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2624,\ (\ :0),\ 0.927 - 0.374i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.482665412\)
\(L(\frac12)\) \(\approx\) \(1.482665412\)
\(L(1)\) 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 \( 1 + (-0.453 + 0.891i)T \)
good3 \( 1 + (-0.399 - 0.965i)T + (-0.707 + 0.707i)T^{2} \)
5 \( 1 + (-0.587 - 0.809i)T^{2} \)
7 \( 1 + (-0.453 - 0.891i)T^{2} \)
11 \( 1 + (0.101 + 1.29i)T + (-0.987 + 0.156i)T^{2} \)
13 \( 1 + (-0.891 - 0.453i)T^{2} \)
17 \( 1 + (-1.20 + 1.40i)T + (-0.156 - 0.987i)T^{2} \)
19 \( 1 + (-0.108 - 0.453i)T + (-0.891 + 0.453i)T^{2} \)
23 \( 1 + (-0.309 - 0.951i)T^{2} \)
29 \( 1 + (0.156 - 0.987i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 + (1.16 - 0.183i)T + (0.951 - 0.309i)T^{2} \)
47 \( 1 + (-0.453 + 0.891i)T^{2} \)
53 \( 1 + (-0.156 + 0.987i)T^{2} \)
59 \( 1 + (0.831 - 1.14i)T + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.951 - 0.309i)T^{2} \)
67 \( 1 + (-0.156 - 0.0123i)T + (0.987 + 0.156i)T^{2} \)
71 \( 1 + (-0.987 + 0.156i)T^{2} \)
73 \( 1 + (1.14 + 1.14i)T + iT^{2} \)
79 \( 1 + (-0.707 + 0.707i)T^{2} \)
83 \( 1 - 1.90iT - T^{2} \)
89 \( 1 + (0.0819 - 0.133i)T + (-0.453 - 0.891i)T^{2} \)
97 \( 1 + (1.93 + 0.152i)T + (0.987 + 0.156i)T^{2} \)
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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

−9.203072743440518138887526007828, −8.507531320780498173514413020385, −7.67998456749645083456962977030, −6.89045299553302816675092375709, −5.77320107284052265548028558810, −5.21420761533637891819840196925, −4.25376530439501010148623573713, −3.32281765386735920753420750366, −2.90609903313772532438594102018, −1.13272427363044811115347449746, 1.34875312185986664693987292299, 2.12812078451391936026980986601, 3.13413058094333412910487749683, 4.26721823055996841513182428758, 5.06688003598716485793503502517, 6.14168582467427683466627828071, 6.85474039888710216935983647229, 7.51492453803589656669690100370, 8.131321743566254773792403215264, 8.761654919774296877487264617088

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