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Properties

Label 2-637-13.9-c1-0-28
Degree $2$
Conductor $637$
Sign $0.566 + 0.824i$
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  + (−0.5 + 0.866i)2-s + (0.707 − 1.22i)3-s + (0.500 + 0.866i)4-s − 4.09·5-s + (0.707 + 1.22i)6-s − 3·8-s + (0.500 + 0.866i)9-s + (2.04 − 3.54i)10-s + (1.89 − 3.28i)11-s + 1.41·12-s + (0.634 − 3.54i)13-s + (−2.89 + 5.01i)15-s + (0.500 − 0.866i)16-s + (−0.634 − 1.09i)17-s − 1.00·18-s + (−1.41 − 2.44i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.408 − 0.707i)3-s + (0.250 + 0.433i)4-s − 1.83·5-s + (0.288 + 0.499i)6-s − 1.06·8-s + (0.166 + 0.288i)9-s + (0.647 − 1.12i)10-s + (0.572 − 0.991i)11-s + 0.408·12-s + (0.176 − 0.984i)13-s + (−0.748 + 1.29i)15-s + (0.125 − 0.216i)16-s + (−0.153 − 0.266i)17-s − 0.235·18-s + (−0.324 − 0.561i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.566 + 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.566 + 0.824i$
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: \((2,\ 637,\ (\ :1/2),\ 0.566 + 0.824i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.756708 - 0.398206i\)
\(L(\frac12)\) \(\approx\) \(0.756708 - 0.398206i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.634 + 3.54i)T \)
good2 \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.707 + 1.22i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 4.09T + 5T^{2} \)
11 \( 1 + (-1.89 + 3.28i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.634 + 1.09i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.41 + 2.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.89 + 6.75i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.397 - 0.689i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.41T + 31T^{2} \)
37 \( 1 + (1.39 - 2.42i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.48 + 2.57i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.89 + 6.75i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + 12.5T + 53T^{2} \)
59 \( 1 + (6.21 + 10.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.17 - 7.22i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.89 + 3.28i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 12.5T + 73T^{2} \)
79 \( 1 + 2.20T + 79T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 + (7.48 - 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.12 + 3.67i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.78179678117143708607293338192, −8.981241484820736069845600791414, −8.262170893862692574803118450506, −8.008515093698000955842782518214, −7.04173194717886273713438738542, −6.51887812640304525106412077395, −4.85820613259197606235553040773, −3.57910202067471369815569415453, −2.82150506863908742318423780555, −0.53232870906208110014845274271, 1.45609047296740185031699039074, 3.18876440264667660329093342746, 3.97218738204525399490237577974, 4.74355490065593425920397134226, 6.46499156511041531122446872036, 7.23831406617797693454834055276, 8.324559688547476036254226184074, 9.251734122052641190809261776357, 9.721405955812685335685363952160, 10.82729759026282260547301717660

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