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Properties

Label 2-637-91.9-c1-0-34
Degree $2$
Conductor $637$
Sign $-0.403 + 0.914i$
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  + (−0.5 − 0.866i)2-s + 1.41·3-s + (0.500 − 0.866i)4-s + (1.34 − 2.32i)5-s + (−0.707 − 1.22i)6-s − 3·8-s − 0.999·9-s − 2.68·10-s + 5.79·11-s + (0.707 − 1.22i)12-s + (2.75 − 2.32i)13-s + (1.89 − 3.28i)15-s + (0.500 + 0.866i)16-s + (−2.75 + 4.77i)17-s + (0.499 + 0.866i)18-s − 2.82·19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + 0.816·3-s + (0.250 − 0.433i)4-s + (0.600 − 1.03i)5-s + (−0.288 − 0.499i)6-s − 1.06·8-s − 0.333·9-s − 0.848·10-s + 1.74·11-s + (0.204 − 0.353i)12-s + (0.764 − 0.644i)13-s + (0.490 − 0.848i)15-s + (0.125 + 0.216i)16-s + (−0.668 + 1.15i)17-s + (0.117 + 0.204i)18-s − 0.648·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.403 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.403 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.403 + 0.914i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.403 + 0.914i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03192 - 1.58301i\)
\(L(\frac12)\) \(\approx\) \(1.03192 - 1.58301i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.75 + 2.32i)T \)
good2 \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 - 1.41T + 3T^{2} \)
5 \( 1 + (-1.34 + 2.32i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 5.79T + 11T^{2} \)
17 \( 1 + (2.75 - 4.77i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 + (0.897 + 1.55i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.39 + 7.61i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.707 - 1.22i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.39 - 5.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.87 - 8.44i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.897 - 1.55i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.41 - 2.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.29 + 5.70i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.562 - 0.974i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 1.55T + 61T^{2} \)
67 \( 1 - 5.79T + 67T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.90 - 5.02i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.89 + 10.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.89T + 83T^{2} \)
89 \( 1 + (6.07 + 10.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.12 - 3.67i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.07109131279886012776623395284, −9.441524482170353099736927759621, −8.590664189129225739887004615568, −8.362088852021701826126259541693, −6.33453370061967938334141287606, −6.08834094245439298500169015516, −4.54980510419738642388583022399, −3.38715365784176462377219290419, −2.06482050693355349284916483036, −1.14491664903449960956496224660, 2.09584880731816063498877028145, 3.10467320567318423440422779689, 3.99739872986276772037448628309, 5.87707108338576436880143467347, 6.75566509008378531539799466136, 7.04419614556857134867049762351, 8.420324923519245003416176306454, 8.980039289664216344149616489111, 9.551872070958393562133018880678, 10.94694122971177520253476109196

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