↑

Properties

Label 2-637-13.10-c1-0-36
Degree $2$
Conductor $637$
Sign $0.746 + 0.665i$
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  + (1.82 − 1.05i)2-s + (1.13 + 1.95i)3-s + (1.22 − 2.12i)4-s − 3.60i·5-s + (4.13 + 2.38i)6-s − 0.948i·8-s + (−1.05 + 1.83i)9-s + (−3.79 − 6.57i)10-s + (0.767 − 0.443i)11-s + 5.53·12-s + (1.17 + 3.40i)13-s + (7.05 − 4.07i)15-s + (1.44 + 2.51i)16-s + (2.48 − 4.29i)17-s + 4.46i·18-s + (−2.06 − 1.18i)19-s + ⋯
L(s)  = 1  + (1.29 − 0.745i)2-s + (0.652 + 1.13i)3-s + (0.612 − 1.06i)4-s − 1.61i·5-s + (1.68 + 0.973i)6-s − 0.335i·8-s + (−0.352 + 0.610i)9-s + (−1.20 − 2.08i)10-s + (0.231 − 0.133i)11-s + 1.59·12-s + (0.325 + 0.945i)13-s + (1.82 − 1.05i)15-s + (0.362 + 0.627i)16-s + (0.601 − 1.04i)17-s + 1.05i·18-s + (−0.472 − 0.272i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.746 + 0.665i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.746 + 0.665i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.35712 - 1.27997i\)
\(L(\frac12)\) \(\approx\) \(3.35712 - 1.27997i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.17 - 3.40i)T \)
good2 \( 1 + (-1.82 + 1.05i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.13 - 1.95i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 3.60iT - 5T^{2} \)
11 \( 1 + (-0.767 + 0.443i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.48 + 4.29i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.06 + 1.18i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.92 + 3.34i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.640 + 1.11i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.46iT - 31T^{2} \)
37 \( 1 + (8.34 - 4.81i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (10.4 - 6.04i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.82 - 3.15i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.98iT - 47T^{2} \)
53 \( 1 - 4.92T + 53T^{2} \)
59 \( 1 + (6.34 + 3.66i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.769 - 1.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.29 + 4.21i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.58 + 3.22i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 7.14iT - 73T^{2} \)
79 \( 1 - 0.757T + 79T^{2} \)
83 \( 1 + 4.76iT - 83T^{2} \)
89 \( 1 + (3.13 - 1.80i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.401 - 0.231i)T + (48.5 + 84.0i)T^{2} \)
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

−10.49642453678852249225734203570, −9.693214051586658785259398966851, −8.793545125154132635156792622204, −8.389712274062774278152287203899, −6.56225381909505565796221812950, −5.13237793641771882069361665742, −4.79115935247672711504733342469, −3.98083032021252549332346789544, −3.10115204611854944731785869544, −1.59237694314722811361901355055, 2.01582090993727595662012622045, 3.22168359758481829128648887248, 3.82622657711107712840756253225, 5.59257717507722544108016889875, 6.24315549202140033528987585474, 7.07349068977170676652495006560, 7.57024467029746189353714766707, 8.398616147077570912337502458075, 10.01771379614995078389522551806, 10.73442072517114458076128386404

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