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Properties

Label 2-2627-2627.780-c0-0-4
Degree $2$
Conductor $2627$
Sign $0.762 + 0.646i$
Analytic cond. $1.31104$
Root an. cond. $1.14500$
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  + (−1.00 + 1.19i)2-s + (1.48 − 1.24i)3-s + (−0.250 − 1.42i)4-s + (0.0340 − 0.0936i)5-s + 3.03i·6-s + (0.602 + 0.347i)8-s + (0.478 − 2.71i)9-s + (0.0779 + 0.134i)10-s + (−2.14 − 1.80i)12-s + (−0.0660 − 0.181i)15-s + (0.335 − 0.122i)16-s + (2.77 + 3.30i)18-s + (−0.963 − 1.14i)19-s + (−0.141 − 0.0250i)20-s + (1.32 − 0.234i)24-s + (0.758 + 0.636i)25-s + ⋯
L(s)  = 1  + (−1.00 + 1.19i)2-s + (1.48 − 1.24i)3-s + (−0.250 − 1.42i)4-s + (0.0340 − 0.0936i)5-s + 3.03i·6-s + (0.602 + 0.347i)8-s + (0.478 − 2.71i)9-s + (0.0779 + 0.134i)10-s + (−2.14 − 1.80i)12-s + (−0.0660 − 0.181i)15-s + (0.335 − 0.122i)16-s + (2.77 + 3.30i)18-s + (−0.963 − 1.14i)19-s + (−0.141 − 0.0250i)20-s + (1.32 − 0.234i)24-s + (0.758 + 0.636i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2627\)    =    \(37 \cdot 71\)
Sign: $0.762 + 0.646i$
Analytic conductor: \(1.31104\)
Root analytic conductor: \(1.14500\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2627} (780, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2627,\ (\ :0),\ 0.762 + 0.646i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.104927222\)
\(L(\frac12)\) \(\approx\) \(1.104927222\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-0.661 - 0.749i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
good2 \( 1 + (1.00 - 1.19i)T + (-0.173 - 0.984i)T^{2} \)
3 \( 1 + (-1.48 + 1.24i)T + (0.173 - 0.984i)T^{2} \)
5 \( 1 + (-0.0340 + 0.0936i)T + (-0.766 - 0.642i)T^{2} \)
7 \( 1 + (-0.766 - 0.642i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.939 - 0.342i)T^{2} \)
19 \( 1 + (0.963 + 1.14i)T + (-0.173 + 0.984i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.975 + 0.563i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
41 \( 1 + (0.939 - 0.342i)T^{2} \)
43 \( 1 + 1.98iT - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (0.766 - 0.642i)T^{2} \)
61 \( 1 + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 - 1.08T + T^{2} \)
79 \( 1 + (0.574 - 1.57i)T + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (-0.126 + 0.719i)T + (-0.939 - 0.342i)T^{2} \)
89 \( 1 + (-0.658 - 1.80i)T + (-0.766 + 0.642i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−8.780550774232144028114162340846, −8.197128691815421640703969913882, −7.49940762901036757485588031978, −6.95913689647184640951084747970, −6.46150364288734960653153874459, −5.50424126726709330572966996400, −4.08155670701260856332686920542, −2.98213137783135268260957454630, −2.04371592566285448718437488242, −0.849175243010357357518409731273, 1.72550918597943225928196660563, 2.47547846045582100589400582790, 3.27463357368729994994734030132, 3.93385081086895208676962435985, 4.74674161899424329255732001405, 5.98008188791998736327816858099, 7.47580624809926856135981464061, 8.091950894791060676822802090397, 8.698099600044948081147354501673, 9.218290756076399416942027529525

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