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Properties

Label 2-637-13.3-c1-0-14
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

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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: $\chi_{637} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.384 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50939 + 2.26506i\)
\(L(\frac12)\) \(\approx\) \(1.50939 + 2.26506i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.486 + 0.843i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.59642131982415613345034539959, −10.05005414095369143970241146965, −8.936365249817399328127448105663, −8.336257165473213422778812221928, −7.21554929619358064909680851743, −6.10832746671092656846130164387, −5.55766558227228067816851031032, −4.18162897725555001876742784698, −3.80972534675549788431734296624, −2.01061848449569760489930310482, 1.42568431320038204824636494391, 2.35079628947152365897669539918, 3.26574645031640555276931114328, 4.46254209251003209617155633425, 5.67747042755073640721513107154, 7.01551029957082355296052949411, 7.54022606915036752716598066733, 8.511427369344485678857459171185, 9.492894849439807616682533248596, 10.60032147713254548156925799165

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