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Properties

Label 2-637-13.3-c1-0-35
Degree $2$
Conductor $637$
Sign $0.0910 + 0.995i$
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 + 2.68·5-s + (0.707 − 1.22i)6-s − 3·8-s + (0.500 − 0.866i)9-s + (−1.34 − 2.32i)10-s + (−2.89 − 5.01i)11-s + 1.41·12-s + (−2.75 − 2.32i)13-s + (1.89 + 3.28i)15-s + (0.500 + 0.866i)16-s + (2.75 − 4.77i)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 + 1.20·5-s + (0.288 − 0.499i)6-s − 1.06·8-s + (0.166 − 0.288i)9-s + (−0.424 − 0.735i)10-s + (−0.873 − 1.51i)11-s + 0.408·12-s + (−0.764 − 0.644i)13-s + (0.490 + 0.848i)15-s + (0.125 + 0.216i)16-s + (0.668 − 1.15i)17-s − 0.235·18-s + (−0.324 + 0.561i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0910 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0910 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.0910 + 0.995i$
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: \((2,\ 637,\ (\ :1/2),\ 0.0910 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23445 - 1.12671i\)
\(L(\frac12)\) \(\approx\) \(1.23445 - 1.12671i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.75 + 2.32i)T \)
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 \( 1 - 2.68T + 5T^{2} \)
11 \( 1 + (2.89 + 5.01i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.75 + 4.77i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.41 - 2.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.897 + 1.55i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.39 - 7.61i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.41T + 31T^{2} \)
37 \( 1 + (-3.39 - 5.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.87 - 8.44i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.897 + 1.55i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 - 6.59T + 53T^{2} \)
59 \( 1 + (-0.562 + 0.974i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.779 + 1.34i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.89 + 5.01i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 5.80T + 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 + (-6.07 - 10.5i)T + (-44.5 + 77.0i)T^{2} \)
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

−10.19412828352523311799795104499, −9.786578984645920463775055461304, −9.008823285113054234305765720158, −8.080714682681255716862561081337, −6.59081439986157425309242539116, −5.72702079488706144536748750665, −4.99096804346814501108726176686, −3.15993074784555366260208913196, −2.65024334726365737682228740939, −0.977760354114844897107363975100, 2.05312526508072952704541987436, 2.46150903536508792741100307748, 4.38238361967451160998811537737, 5.63698870430492505635840110897, 6.55759822120836391596107954792, 7.40859353588782133375642776116, 7.85669017385208443750199452557, 8.930681543370470385883079646928, 9.835032416503586328604294107453, 10.41882806740735312595596705281

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