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Properties

Label 2-637-13.3-c1-0-25
Degree $2$
Conductor $637$
Sign $-0.0128 + 0.999i$
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  + (−0.190 − 0.330i)2-s + (−0.190 − 0.330i)3-s + (0.927 − 1.60i)4-s − 0.381·5-s + (−0.0729 + 0.126i)6-s − 1.47·8-s + (1.42 − 2.47i)9-s + (0.0729 + 0.126i)10-s + (2.42 + 4.20i)11-s − 0.708·12-s + (2.5 + 2.59i)13-s + (0.0729 + 0.126i)15-s + (−1.57 − 2.72i)16-s + (3.73 − 6.47i)17-s − 1.09·18-s + (2.42 − 4.20i)19-s + ⋯
L(s)  = 1  + (−0.135 − 0.233i)2-s + (−0.110 − 0.190i)3-s + (0.463 − 0.802i)4-s − 0.170·5-s + (−0.0297 + 0.0515i)6-s − 0.520·8-s + (0.475 − 0.823i)9-s + (0.0230 + 0.0399i)10-s + (0.731 + 1.26i)11-s − 0.204·12-s + (0.693 + 0.720i)13-s + (0.0188 + 0.0326i)15-s + (−0.393 − 0.681i)16-s + (0.906 − 1.56i)17-s − 0.256·18-s + (0.556 − 0.964i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.0128 + 0.999i$
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: \((2,\ 637,\ (\ :1/2),\ -0.0128 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05777 - 1.07143i\)
\(L(\frac12)\) \(\approx\) \(1.05777 - 1.07143i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 2.59i)T \)
good2 \( 1 + (0.190 + 0.330i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.190 + 0.330i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 0.381T + 5T^{2} \)
11 \( 1 + (-2.42 - 4.20i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.73 + 6.47i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.42 + 4.20i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.23 + 3.87i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.04 - 3.54i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.70T + 31T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.61 - 4.53i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.78 + 6.54i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.23T + 47T^{2} \)
53 \( 1 - 8.23T + 53T^{2} \)
59 \( 1 + (-1.11 + 1.93i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.354 + 0.613i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.09 - 7.08i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 6.70T + 83T^{2} \)
89 \( 1 + (-8.04 - 13.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.07 + 10.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.27384125135878331664750741728, −9.438858636987416894295558914285, −9.099349094916214775825107146972, −7.25932534373386590949170932877, −6.99377040564688903640473413002, −5.94126395408244912607433308752, −4.82958074799276302840806630308, −3.67536858105666630378441493706, −2.15796666637193059853554129844, −0.953407325176680439080387772387, 1.66834253220644276771174324932, 3.47392850235825359097299150108, 3.84411349974681279307096852970, 5.70128077809935707480892536346, 6.12868206129805023025959856606, 7.66298446917750806771065471390, 7.938500147402844994499809931470, 8.832849789359793443419043317461, 10.05086577533890342274517472785, 10.85055317394487702840883931630

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