Properties

Label 2-637-7.2-c1-0-7
Degree $2$
Conductor $637$
Sign $0.605 - 0.795i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.986418 + 0.488994i\)
\(L(\frac12)\) \(\approx\) \(0.986418 + 0.488994i\)
\(L(\frac{3}{2})\) 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

−11.10746955412724924006116828589, −9.794251117993925509644333827661, −8.720473208440529915708843276356, −7.74682007683468155012351747310, −7.15483767952491840112803450293, −6.50036349460008867071216105659, −5.62850828515683812886938397045, −4.08616416955677113954172633773, −2.80551532454422095122997885018, −1.39682197570714546692433927274, 0.75230714858177226726302529098, 2.57692686102129289810564038069, 4.02051541233478421603970015196, 4.89507096451576081807615852798, 5.90366341639644609967063000181, 6.48400325515701686641066707702, 7.998230508727277926421203902083, 9.044462578526515083663414115452, 9.865854616790587797705492040034, 10.43970087174889203232044648567

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