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Properties

Label 2-637-91.51-c1-0-2
Degree $2$
Conductor $637$
Sign $0.805 - 0.592i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.805 - 0.592i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.805 - 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.222295 + 0.0728730i\)
\(L(\frac12)\) \(\approx\) \(0.222295 + 0.0728730i\)
\(L(\frac{3}{2})\) 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

−10.49352175276716041285100483230, −9.795425368206420947395434537571, −9.058888894922869220921364292794, −7.992587679360477991603418900477, −7.67264209921175912831277713955, −6.34866017776290604503727695000, −4.96672642830003722511841037287, −4.10944822711233307152791515753, −2.54210843681061690195040549495, −1.06412539532344239366578673530, 0.22131153908574293322937716255, 2.61587097991988216020518800517, 3.95948774021449791318145601946, 4.90444285905273747824244333483, 6.40031301759592371282883441750, 7.22343311379498693693868703764, 8.038306012096145070092155260172, 8.301148783574526971724336958469, 9.738855186858660706617698768813, 10.37110630559587417522258670977

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