↑

Properties

Label 2-637-49.43-c1-0-11
Degree $2$
Conductor $637$
Sign $0.332 - 0.943i$
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  + (0.258 − 0.324i)2-s + (−2.07 − 0.998i)3-s + (0.406 + 1.78i)4-s + (2.95 + 1.42i)5-s + (−0.860 + 0.414i)6-s + (2.39 + 1.13i)7-s + (1.43 + 0.689i)8-s + (1.43 + 1.79i)9-s + (1.22 − 0.590i)10-s + (−3.15 + 3.95i)11-s + (0.935 − 4.09i)12-s + (−0.623 + 0.781i)13-s + (0.986 − 0.483i)14-s + (−4.70 − 5.89i)15-s + (−2.69 + 1.30i)16-s + (0.170 − 0.749i)17-s + ⋯
L(s)  = 1  + (0.182 − 0.229i)2-s + (−1.19 − 0.576i)3-s + (0.203 + 0.890i)4-s + (1.32 + 0.636i)5-s + (−0.351 + 0.169i)6-s + (0.903 + 0.427i)7-s + (0.506 + 0.243i)8-s + (0.476 + 0.597i)9-s + (0.387 − 0.186i)10-s + (−0.950 + 1.19i)11-s + (0.270 − 1.18i)12-s + (−0.172 + 0.216i)13-s + (0.263 − 0.129i)14-s + (−1.21 − 1.52i)15-s + (−0.674 + 0.325i)16-s + (0.0414 − 0.181i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.332 - 0.943i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.332 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11000 + 0.785616i\)
\(L(\frac12)\) \(\approx\) \(1.11000 + 0.785616i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.39 - 1.13i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
good2 \( 1 + (-0.258 + 0.324i)T + (-0.445 - 1.94i)T^{2} \)
3 \( 1 + (2.07 + 0.998i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (-2.95 - 1.42i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (3.15 - 3.95i)T + (-2.44 - 10.7i)T^{2} \)
17 \( 1 + (-0.170 + 0.749i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 6.19T + 19T^{2} \)
23 \( 1 + (1.18 + 5.19i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (1.36 - 5.96i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 - 6.89T + 31T^{2} \)
37 \( 1 + (-0.332 + 1.45i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-2.62 - 1.26i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (0.824 - 0.397i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-7.89 + 9.90i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-1.30 - 5.71i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (4.68 - 2.25i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-2.78 + 12.1i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + 4.44T + 67T^{2} \)
71 \( 1 + (0.427 + 1.87i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-5.56 - 6.98i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + 4.91T + 79T^{2} \)
83 \( 1 + (-6.11 - 7.67i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-8.82 - 11.0i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 14.7T + 97T^{2} \)
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

−10.68101433650603647015681226126, −10.42563258735035530125639772327, −9.003427499735970014270470140365, −7.926948215939764230203120396749, −6.96886878054306292800590985463, −6.35746075700842715061861425931, −5.29267924224378197934964272722, −4.52623160188865338216542570259, −2.51151077142178764436661513729, −1.96024291894315455876234568330, 0.795653734736063549366492461728, 2.16240612918128184971958377145, 4.44275907710748163856789171541, 5.08252646227663916422971761556, 5.97791068529818905882711898949, 6.04186375960175134617702621995, 7.72444319643421553199722220724, 8.813748966817502417842675011905, 9.970578297506163347732786536509, 10.42339570708481658993534189538

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