↑

Properties

Label 2-637-13.9-c1-0-11
Degree $2$
Conductor $637$
Sign $0.384 - 0.922i$
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  + (0.760 − 1.31i)2-s + (−1.06 + 1.84i)3-s + (−0.156 − 0.270i)4-s − 0.589·5-s + (1.61 + 2.80i)6-s + 2.56·8-s + (−0.760 − 1.31i)9-s + (−0.448 + 0.776i)10-s + (−0.760 + 1.31i)11-s + 0.664·12-s + (−3.32 + 1.39i)13-s + (0.626 − 1.08i)15-s + (2.26 − 3.92i)16-s + (2.39 + 4.15i)17-s − 2.31·18-s + (0.841 + 1.45i)19-s + ⋯
L(s)  = 1  + (0.537 − 0.931i)2-s + (−0.613 + 1.06i)3-s + (−0.0781 − 0.135i)4-s − 0.263·5-s + (0.660 + 1.14i)6-s + 0.907·8-s + (−0.253 − 0.439i)9-s + (−0.141 + 0.245i)10-s + (−0.229 + 0.397i)11-s + 0.191·12-s + (−0.922 + 0.386i)13-s + (0.161 − 0.280i)15-s + (0.565 − 0.980i)16-s + (0.581 + 1.00i)17-s − 0.545·18-s + (0.193 + 0.334i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.384 - 0.922i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.384 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19100 + 0.793666i\)
\(L(\frac12)\) \(\approx\) \(1.19100 + 0.793666i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.32 - 1.39i)T \)
good2 \( 1 + (-0.760 + 1.31i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.06 - 1.84i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 0.589T + 5T^{2} \)
11 \( 1 + (0.760 - 1.31i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.39 - 4.15i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.841 - 1.45i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.886 - 1.53i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.44 - 5.96i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.08T + 31T^{2} \)
37 \( 1 + (0.704 - 1.22i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.677 + 1.17i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.77 - 10.0i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.464T + 47T^{2} \)
53 \( 1 - 8.24T + 53T^{2} \)
59 \( 1 + (5.93 + 10.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.24 - 2.14i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.78 + 6.55i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.30 + 5.71i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 16.3T + 73T^{2} \)
79 \( 1 + 14.9T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + (-8.24 + 14.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.486 + 0.843i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.83792271760549351377683655653, −10.07530471665828623844191563603, −9.590834576514112787267966791358, −8.072086312271248416643421007843, −7.28766277332740391763409173554, −5.83117510110445585152383349944, −4.85252188583718203242907420173, −4.18959726849041110709906162794, −3.29057157443323716982563814079, −1.87272041631664334366659867596, 0.71053457165757075059635209742, 2.40507548721078497229187371039, 4.14251011538571415330961171129, 5.36054392558397897088730336285, 5.85667906557422222491843185524, 6.90745429644619070007091050113, 7.44585344208448070796575038553, 8.098082563735665238727703414560, 9.586035055152074136869054940301, 10.49876496323365772648126235188

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