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Properties

Label 2-637-49.37-c1-0-25
Degree $2$
Conductor $637$
Sign $0.464 - 0.885i$
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  + (−0.296 + 0.202i)2-s + (1.23 − 1.14i)3-s + (−0.683 + 1.74i)4-s + (1.06 + 0.328i)5-s + (−0.134 + 0.588i)6-s + (2.51 + 0.810i)7-s + (−0.309 − 1.35i)8-s + (−0.0124 + 0.165i)9-s + (−0.382 + 0.118i)10-s + (0.480 + 6.41i)11-s + (1.15 + 2.93i)12-s + (−0.900 − 0.433i)13-s + (−0.910 + 0.268i)14-s + (1.69 − 0.815i)15-s + (−2.37 − 2.20i)16-s + (−6.27 − 0.945i)17-s + ⋯
L(s)  = 1  + (−0.209 + 0.142i)2-s + (0.712 − 0.661i)3-s + (−0.341 + 0.870i)4-s + (0.476 + 0.147i)5-s + (−0.0548 + 0.240i)6-s + (0.951 + 0.306i)7-s + (−0.109 − 0.478i)8-s + (−0.00414 + 0.0552i)9-s + (−0.121 + 0.0373i)10-s + (0.144 + 1.93i)11-s + (0.332 + 0.846i)12-s + (−0.249 − 0.120i)13-s + (−0.243 + 0.0718i)14-s + (0.437 − 0.210i)15-s + (−0.594 − 0.551i)16-s + (−1.52 − 0.229i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.464 - 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.464 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.464 - 0.885i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.464 - 0.885i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47548 + 0.891673i\)
\(L(\frac12)\) \(\approx\) \(1.47548 + 0.891673i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.51 - 0.810i)T \)
13 \( 1 + (0.900 + 0.433i)T \)
good2 \( 1 + (0.296 - 0.202i)T + (0.730 - 1.86i)T^{2} \)
3 \( 1 + (-1.23 + 1.14i)T + (0.224 - 2.99i)T^{2} \)
5 \( 1 + (-1.06 - 0.328i)T + (4.13 + 2.81i)T^{2} \)
11 \( 1 + (-0.480 - 6.41i)T + (-10.8 + 1.63i)T^{2} \)
17 \( 1 + (6.27 + 0.945i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (0.779 - 1.35i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.97 + 0.900i)T + (21.9 - 6.77i)T^{2} \)
29 \( 1 + (-2.37 + 2.97i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (-2.08 - 3.60i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.54 + 9.03i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (-1.28 - 5.62i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (1.79 - 7.85i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-7.59 + 5.18i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (-3.49 + 8.89i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (-3.91 + 1.20i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (-1.70 - 4.33i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (0.455 + 0.788i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.92 + 8.68i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-6.64 - 4.52i)T + (26.6 + 67.9i)T^{2} \)
79 \( 1 + (-3.13 + 5.43i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.0 + 5.33i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-0.125 + 1.67i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 + 12.3T + 97T^{2} \)
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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

−10.65674767505220666232758767078, −9.571431260567607178912596577069, −8.830187327825413001047007946946, −8.101888020717401411990958357679, −7.30374447403716833530659664615, −6.74474819962831681153287573962, −4.98888379048937214276421713467, −4.30260872202481736798438810237, −2.57263209702375270125127929052, −1.95537717312157279090977306964, 0.992218702551996676282479913719, 2.48100037665962453332937449359, 3.86110589133549198325039370275, 4.84272860738513673183723731675, 5.74279137521564873611294403011, 6.75680637100046934324163845994, 8.357034704251019694456724120338, 8.836325645767532669866954212479, 9.345574000442632172516459508487, 10.52210615746530622634088569821

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