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Properties

Label 2-637-91.25-c1-0-17
Degree $2$
Conductor $637$
Sign $0.874 - 0.485i$
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  + (1.43 − 0.827i)2-s + (0.247 − 0.428i)3-s + (0.370 − 0.641i)4-s + (−2.48 + 1.43i)5-s − 0.819i·6-s + 2.08i·8-s + (1.37 + 2.38i)9-s + (−2.37 + 4.12i)10-s + (2.23 + 1.29i)11-s + (−0.183 − 0.317i)12-s + (3.54 − 0.654i)13-s + 1.42i·15-s + (2.46 + 4.27i)16-s + (−1.13 + 1.97i)17-s + (3.94 + 2.28i)18-s + (−3.80 + 2.19i)19-s + ⋯
L(s)  = 1  + (1.01 − 0.585i)2-s + (0.142 − 0.247i)3-s + (0.185 − 0.320i)4-s + (−1.11 + 0.642i)5-s − 0.334i·6-s + 0.737i·8-s + (0.459 + 0.795i)9-s + (−0.752 + 1.30i)10-s + (0.674 + 0.389i)11-s + (−0.0529 − 0.0916i)12-s + (0.983 − 0.181i)13-s + 0.367i·15-s + (0.616 + 1.06i)16-s + (−0.275 + 0.478i)17-s + (0.930 + 0.537i)18-s + (−0.873 + 0.504i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.874 - 0.485i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.874 - 0.485i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.08707 + 0.540494i\)
\(L(\frac12)\) \(\approx\) \(2.08707 + 0.540494i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.54 + 0.654i)T \)
good2 \( 1 + (-1.43 + 0.827i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.247 + 0.428i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.48 - 1.43i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.23 - 1.29i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.13 - 1.97i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.80 - 2.19i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.58 + 6.20i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.19T + 29T^{2} \)
31 \( 1 + (-5.93 - 3.42i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.10 - 2.94i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.98iT - 41T^{2} \)
43 \( 1 - 1.31T + 43T^{2} \)
47 \( 1 + (1.17 - 0.676i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.14 + 7.17i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (8.32 + 4.80i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.77 - 8.27i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.7 + 7.95i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.79iT - 71T^{2} \)
73 \( 1 + (-4.94 - 2.85i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.00 - 1.74i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.11iT - 83T^{2} \)
89 \( 1 + (-11.3 + 6.52i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 14.0iT - 97T^{2} \)
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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

−10.70131038734974664915709297042, −10.39771176600588856471353951929, −8.437697186718022151628849995424, −8.252010171616444721686478150463, −6.98102708128024987053199806717, −6.14968039379142931893265360750, −4.61203764078434467153743861759, −4.09138646154110496745669158285, −3.12097079934097183842703938964, −1.88909463783163505887503518627, 0.920585213722609911931395357043, 3.41556321901363483068091306893, 4.11011462435483571158489456530, 4.68767664543462208477847536147, 6.04808016020453550621021559295, 6.66856128846401009160777133975, 7.75471529164883859644914668651, 8.761975681995738510570735641192, 9.421243759275415577774345641779, 10.58767325865454934958978178442

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