Properties

Label 2-637-49.4-c1-0-42
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

Related objects

Downloads

Learn more

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: $\chi_{637} (53, \cdot )$
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 \( 1 + 12.3T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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