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Properties

Label 2-637-91.19-c1-0-2
Degree $2$
Conductor $637$
Sign $-0.597 - 0.802i$
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  + (−1.56 − 0.419i)2-s − 0.513i·3-s + (0.546 + 0.315i)4-s + (−1.47 + 0.395i)5-s + (−0.215 + 0.805i)6-s + (1.57 + 1.57i)8-s + 2.73·9-s + 2.47·10-s + (−2.03 − 2.03i)11-s + (0.162 − 0.280i)12-s + (−2.94 + 2.08i)13-s + (0.203 + 0.757i)15-s + (−2.43 − 4.21i)16-s + (3.10 − 5.38i)17-s + (−4.28 − 1.14i)18-s + (−1.63 − 1.63i)19-s + ⋯
L(s)  = 1  + (−1.10 − 0.296i)2-s − 0.296i·3-s + (0.273 + 0.157i)4-s + (−0.659 + 0.176i)5-s + (−0.0880 + 0.328i)6-s + (0.555 + 0.555i)8-s + 0.911·9-s + 0.783·10-s + (−0.614 − 0.614i)11-s + (0.0467 − 0.0810i)12-s + (−0.816 + 0.577i)13-s + (0.0524 + 0.195i)15-s + (−0.607 − 1.05i)16-s + (0.753 − 1.30i)17-s + (−1.01 − 0.270i)18-s + (−0.375 − 0.375i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.597 - 0.802i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.597 - 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0349159 + 0.0695171i\)
\(L(\frac12)\) \(\approx\) \(0.0349159 + 0.0695171i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.94 - 2.08i)T \)
good2 \( 1 + (1.56 + 0.419i)T + (1.73 + i)T^{2} \)
3 \( 1 + 0.513iT - 3T^{2} \)
5 \( 1 + (1.47 - 0.395i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (2.03 + 2.03i)T + 11iT^{2} \)
17 \( 1 + (-3.10 + 5.38i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.63 + 1.63i)T + 19iT^{2} \)
23 \( 1 + (4.86 - 2.81i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.379 + 0.656i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.24 - 8.36i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (1.56 - 5.82i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.94 - 0.522i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.24 - 1.29i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.571 - 2.13i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.47 - 4.28i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.228 + 0.851i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + 5.17iT - 61T^{2} \)
67 \( 1 + (11.1 - 11.1i)T - 67iT^{2} \)
71 \( 1 + (10.3 + 2.76i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (6.72 + 1.80i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.24 - 7.35i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.51 + 1.51i)T + 83iT^{2} \)
89 \( 1 + (-8.08 - 2.16i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.933 - 3.48i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.66010739310522850857774911026, −9.962489259984003978251900196385, −9.276584592666571530692834155492, −8.228483319839766751511820407792, −7.54098087305982846721356852436, −6.94700173086493003325463630816, −5.35340672065590090086201350601, −4.37795563515041769236917687601, −2.89992947676276934506124148782, −1.49906540068117132571792307828, 0.06330259564177129637537793732, 1.89076996496210939362271937258, 3.84098052796546673533570918291, 4.48376580980389963407407718111, 5.86474605309166846685169407474, 7.21589345955115028955488665789, 7.78373196794137041172541318437, 8.356447014622615271649556642319, 9.503465493582817552589050524183, 10.27457663762388080882940754761

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