↑

Properties

Label 2-637-91.23-c1-0-36
Degree $2$
Conductor $637$
Sign $-0.372 + 0.927i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.456i·2-s + (1.39 − 2.41i)3-s + 1.79·4-s + (−0.395 − 0.228i)5-s + (−1.10 − 0.637i)6-s − 1.73i·8-s + (−2.39 − 4.14i)9-s + (−0.104 + 0.180i)10-s + (3.39 + 1.96i)11-s + (2.5 − 4.33i)12-s + (−3.5 − 0.866i)13-s + (−1.10 + 0.637i)15-s + 2.79·16-s − 3·17-s + (−1.89 + 1.09i)18-s + (1.18 − 0.685i)19-s + ⋯
L(s)  = 1  − 0.323i·2-s + (0.805 − 1.39i)3-s + 0.895·4-s + (−0.176 − 0.102i)5-s + (−0.450 − 0.260i)6-s − 0.612i·8-s + (−0.798 − 1.38i)9-s + (−0.0330 + 0.0571i)10-s + (1.02 + 0.591i)11-s + (0.721 − 1.25i)12-s + (−0.970 − 0.240i)13-s + (−0.285 + 0.164i)15-s + 0.697·16-s − 0.727·17-s + (−0.446 + 0.257i)18-s + (0.272 − 0.157i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.372 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.372 + 0.927i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.372 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28034 - 1.89444i\)
\(L(\frac12)\) \(\approx\) \(1.28034 - 1.89444i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.5 + 0.866i)T \)
good2 \( 1 + 0.456iT - 2T^{2} \)
3 \( 1 + (-1.39 + 2.41i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.395 + 0.228i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.39 - 1.96i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + (-1.18 + 0.685i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.58T + 23T^{2} \)
29 \( 1 + (-3.39 - 5.88i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-7.5 + 4.33i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.92iT - 37T^{2} \)
41 \( 1 + (6.79 - 3.92i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.68 - 8.11i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-8.29 - 4.78i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.08 + 5.33i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 12.3iT - 59T^{2} \)
61 \( 1 + (7.37 + 12.7i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.87 - 2.23i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.79 - 2.18i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (3 - 1.73i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3 + 5.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.02iT - 83T^{2} \)
89 \( 1 - 16.1iT - 89T^{2} \)
97 \( 1 + (-6.31 - 3.64i)T + (48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.25456436509819797530968980076, −9.438330609683080154154176879487, −8.296139100345401163773963796824, −7.65603876875911550180234183376, −6.73026388398757019681449655512, −6.39050970461606434915508643514, −4.56559258033460906841430322472, −3.10690378896087616522524560119, −2.26865212919591692733733551697, −1.24420679570262469178396602579, 2.20527056451832008666077908539, 3.29720734956952129651356291656, 4.18422274463419537881382882272, 5.28170885232712290420499610630, 6.42883799761839721886692724428, 7.37334985932552469135494613465, 8.394337084916896903891577095202, 9.088761147536696557379497046712, 9.995010186324360782261770556849, 10.65113141764913839070341471393

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