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Properties

Label 2-637-13.4-c1-0-27
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

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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: $\chi_{637} (589, \cdot )$
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 \( 1 + (-0.401 + 0.231i)T + (48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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