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Properties

Label 2-637-91.74-c1-0-4
Degree $2$
Conductor $637$
Sign $0.788 - 0.615i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
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  + 2-s + (−1.5 − 2.59i)3-s − 4-s + (1.5 + 2.59i)5-s + (−1.5 − 2.59i)6-s − 3·8-s + (−3 + 5.19i)9-s + (1.5 + 2.59i)10-s + (1.5 + 2.59i)11-s + (1.5 + 2.59i)12-s + (1 + 3.46i)13-s + (4.5 − 7.79i)15-s − 16-s + 2·17-s + (−3 + 5.19i)18-s + (−0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.866 − 1.49i)3-s − 0.5·4-s + (0.670 + 1.16i)5-s + (−0.612 − 1.06i)6-s − 1.06·8-s + (−1 + 1.73i)9-s + (0.474 + 0.821i)10-s + (0.452 + 0.783i)11-s + (0.433 + 0.749i)12-s + (0.277 + 0.960i)13-s + (1.16 − 2.01i)15-s − 0.250·16-s + 0.485·17-s + (−0.707 + 1.22i)18-s + (−0.114 + 0.198i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.788 - 0.615i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.788 - 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16211 + 0.400115i\)
\(L(\frac12)\) \(\approx\) \(1.16211 + 0.400115i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 + (-1 - 3.46i)T \)
good2 \( 1 - T + 2T^{2} \)
3 \( 1 + (1.5 + 2.59i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (3.5 - 6.06i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.5 + 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.5 - 6.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + (6.5 - 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.5 + 11.2i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.5 - 11.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.5 - 2.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (2.5 + 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.97362943160390685804829381039, −9.922556701233750088242751248758, −8.949953815024556367006728802541, −7.59423588506133414293067071337, −6.83558622988435071504855307682, −6.19608207106081283978248930988, −5.52199942051984384581974102511, −4.24172625753363698227480771009, −2.75810618453237808642489293905, −1.55535963690720593600409619946, 0.63879991993342906314309035535, 3.28969065644276697809865200440, 4.17980600137660020231207138725, 4.99440595445804030262982022108, 5.64632314701751292668146259107, 6.11055666883225270480791372074, 8.253635428651120014387623823920, 9.069295824135164454064268136349, 9.600246175692059040102002772527, 10.40862877019752126746322511114

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