↑

Properties

Label 2-637-91.4-c1-0-23
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: $\chi_{637} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.372 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29389 - 0.874471i\)
\(L(\frac12)\) \(\approx\) \(1.29389 - 0.874471i\)
\(L(\frac{3}{2})\) 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

−10.93255889481806058243812868331, −9.501379518986480841069363609490, −8.258185678958339729471424206629, −7.60432479327146753986803404896, −6.70849290467544021938648239693, −6.02754290170039811582021070218, −5.58091600014260064667609010277, −3.65128888613961709687098563126, −2.08721821007536886643836102810, −1.06001833414365627601565033938, 1.56922570399140501144258352674, 3.33473671278930713498370274384, 4.06722926122427706905716311439, 5.23009486497388365542472260837, 6.23294361841214697552853423746, 6.82152337185676039283662164618, 8.299408809323139093153817896978, 9.452764038797859625091863957229, 10.06306393533723756994617105063, 10.72434972235414617573360021568

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