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Properties

Label 2-637-7.4-c1-0-33
Degree $2$
Conductor $637$
Sign $-0.701 + 0.712i$
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  + (−0.235 − 0.407i)2-s + (1.12 − 1.94i)3-s + (0.889 − 1.54i)4-s + (0.264 + 0.458i)5-s − 1.05·6-s − 1.77·8-s + (−1.02 − 1.78i)9-s + (0.124 − 0.215i)10-s + (1.12 − 1.94i)11-s + (−2 − 3.46i)12-s − 13-s + 1.19·15-s + (−1.35 − 2.35i)16-s + (−0.653 + 1.13i)17-s + (−0.484 + 0.839i)18-s + (−0.735 − 1.27i)19-s + ⋯
L(s)  = 1  + (−0.166 − 0.288i)2-s + (0.649 − 1.12i)3-s + (0.444 − 0.770i)4-s + (0.118 + 0.205i)5-s − 0.432·6-s − 0.628·8-s + (−0.343 − 0.594i)9-s + (0.0393 − 0.0682i)10-s + (0.339 − 0.587i)11-s + (−0.577 − 0.999i)12-s − 0.277·13-s + 0.307·15-s + (−0.339 − 0.588i)16-s + (−0.158 + 0.274i)17-s + (−0.114 + 0.197i)18-s + (−0.168 − 0.292i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.701 + 0.712i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.694550 - 1.65730i\)
\(L(\frac12)\) \(\approx\) \(0.694550 - 1.65730i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good2 \( 1 + (0.235 + 0.407i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.12 + 1.94i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.264 - 0.458i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.12 + 1.94i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.653 - 1.13i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.735 + 1.27i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.91 + 5.05i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.22T + 29T^{2} \)
31 \( 1 + (3.51 - 6.08i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.18 - 2.04i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.49T + 41T^{2} \)
43 \( 1 - 11.3T + 43T^{2} \)
47 \( 1 + (-4.29 - 7.43i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.63 - 9.76i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.08 - 10.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.96 + 13.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.19T + 71T^{2} \)
73 \( 1 + (-3.82 + 6.61i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.669 - 1.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.3T + 83T^{2} \)
89 \( 1 + (-3.45 - 5.98i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.47T + 97T^{2} \)
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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

−10.49726831870870203270142443153, −9.265363122736429375388088589710, −8.550283833383517438823399445424, −7.59340901745017376195070855378, −6.57539220263257344102685876599, −6.17372415647096101694066208888, −4.70573741139571697393734194802, −2.98731211646972824621625050261, −2.18656468621651464154888870987, −0.983812211928118496442745611831, 2.22097348516636377598838239691, 3.44331498491179379180998787756, 4.15657259153233308153630103206, 5.31206813958636241939650218628, 6.61123358252907738291252610109, 7.53082623204882011513217171530, 8.378355567937506824340497086470, 9.267135889733640635614787932156, 9.715484114102007685538192140072, 10.78444046618435980699977732218

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