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Properties

Label 2-637-91.45-c1-0-11
Degree $2$
Conductor $637$
Sign $0.777 - 0.628i$
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  + (1.67 − 1.67i)2-s + (−1.04 + 0.601i)3-s − 3.60i·4-s + (−0.825 + 3.08i)5-s + (−0.736 + 2.74i)6-s + (−2.68 − 2.68i)8-s + (−0.777 + 1.34i)9-s + (3.77 + 6.54i)10-s + (−1.10 + 4.11i)11-s + (2.16 + 3.75i)12-s + (−3.48 + 0.920i)13-s + (−0.992 − 3.70i)15-s − 1.79·16-s + 1.44·17-s + (0.952 + 3.55i)18-s + (2.42 − 0.649i)19-s + ⋯
L(s)  = 1  + (1.18 − 1.18i)2-s + (−0.601 + 0.347i)3-s − 1.80i·4-s + (−0.369 + 1.37i)5-s + (−0.300 + 1.12i)6-s + (−0.950 − 0.950i)8-s + (−0.259 + 0.448i)9-s + (1.19 + 2.06i)10-s + (−0.332 + 1.23i)11-s + (0.625 + 1.08i)12-s + (−0.966 + 0.255i)13-s + (−0.256 − 0.956i)15-s − 0.448·16-s + 0.350·17-s + (0.224 + 0.838i)18-s + (0.556 − 0.149i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 - 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.777 - 0.628i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.777 - 0.628i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56351 + 0.552853i\)
\(L(\frac12)\) \(\approx\) \(1.56351 + 0.552853i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.48 - 0.920i)T \)
good2 \( 1 + (-1.67 + 1.67i)T - 2iT^{2} \)
3 \( 1 + (1.04 - 0.601i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.825 - 3.08i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.10 - 4.11i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 - 1.44T + 17T^{2} \)
19 \( 1 + (-2.42 + 0.649i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 5.23iT - 23T^{2} \)
29 \( 1 + (-1.34 + 2.32i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.14 - 1.37i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-0.438 - 0.438i)T + 37iT^{2} \)
41 \( 1 + (-5.04 + 1.35i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (5.46 - 3.15i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.39 - 1.71i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.79 + 6.56i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.43 - 1.43i)T - 59iT^{2} \)
61 \( 1 + (-4.53 - 2.62i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.46 + 2.26i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-8.31 - 2.22i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-4.09 - 15.2i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.00 + 1.74i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.11 + 5.11i)T + 83iT^{2} \)
89 \( 1 + (-4.95 + 4.95i)T - 89iT^{2} \)
97 \( 1 + (1.62 - 6.06i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.91757735987629895883861145272, −10.15922207629701001260645137149, −9.691404024770602946701489914518, −7.70492679044192155892247143994, −7.01148738421811878622778551646, −5.70268997339959387796894473502, −4.97460666585071761878355560575, −4.04751328946594960205307457458, −2.95126661543616399458344706313, −2.11848715506967071430541151664, 0.65386073665267414243324273697, 3.18441002807914944531035926775, 4.31253133462708993779397341111, 5.32020361606913189453570316476, 5.64384111661654014417692277834, 6.69604742564682578405473534655, 7.67078438065026987815432732795, 8.397388061965419458914352359780, 9.249913757021719502806236605824, 10.73858987377348180072747160750

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