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Properties

Label 2-637-91.31-c1-0-22
Degree $2$
Conductor $637$
Sign $-0.966 + 0.256i$
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  + (−1.16 − 0.312i)2-s + (−1.96 − 1.13i)3-s + (−0.466 − 0.269i)4-s + (−0.224 + 0.837i)5-s + (1.94 + 1.94i)6-s + (2.17 + 2.17i)8-s + (1.08 + 1.87i)9-s + (0.523 − 0.907i)10-s + (2.53 − 0.678i)11-s + (0.612 + 1.06i)12-s + (−0.104 − 3.60i)13-s + (1.39 − 1.39i)15-s + (−1.31 − 2.27i)16-s + (1.52 − 2.63i)17-s + (−0.678 − 2.53i)18-s + (0.0381 − 0.142i)19-s + ⋯
L(s)  = 1  + (−0.825 − 0.221i)2-s + (−1.13 − 0.656i)3-s + (−0.233 − 0.134i)4-s + (−0.100 + 0.374i)5-s + (0.793 + 0.793i)6-s + (0.767 + 0.767i)8-s + (0.361 + 0.626i)9-s + (0.165 − 0.286i)10-s + (0.763 − 0.204i)11-s + (0.176 + 0.306i)12-s + (−0.0289 − 0.999i)13-s + (0.359 − 0.359i)15-s + (−0.328 − 0.569i)16-s + (0.369 − 0.640i)17-s + (−0.160 − 0.597i)18-s + (0.00875 − 0.0326i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.966 + 0.256i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.966 + 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0444935 - 0.340928i\)
\(L(\frac12)\) \(\approx\) \(0.0444935 - 0.340928i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.104 + 3.60i)T \)
good2 \( 1 + (1.16 + 0.312i)T + (1.73 + i)T^{2} \)
3 \( 1 + (1.96 + 1.13i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.224 - 0.837i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.53 + 0.678i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.52 + 2.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0381 + 0.142i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-5.63 + 3.25i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.78T + 29T^{2} \)
31 \( 1 + (9.25 - 2.47i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (0.741 - 2.76i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.27 + 2.27i)T + 41iT^{2} \)
43 \( 1 + 3.18iT - 43T^{2} \)
47 \( 1 + (7.12 + 1.90i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.71 - 2.97i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.34 + 12.5i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-7.97 + 4.60i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.384 + 1.43i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.10 - 4.10i)T - 71iT^{2} \)
73 \( 1 + (-2.53 - 9.46i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (8.79 + 15.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.5 + 10.5i)T + 83iT^{2} \)
89 \( 1 + (4.63 + 1.24i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-4.44 - 4.44i)T + 97iT^{2} \)
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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

−10.34956982975331232617267713816, −9.336442813567865656112565804088, −8.563733957651975814022755624839, −7.40540456716564209981361894643, −6.79573487909397915536924429006, −5.60766416894385400209023239316, −4.96921015285702677447445400148, −3.27697761898244444471315774153, −1.48327512356204429445271004749, −0.34105290844097236555503847237, 1.35421269756282261001032424080, 3.78843968150839128971863856563, 4.53972124978519011503907251133, 5.48537649984562949017138137417, 6.61720197804968434306137729208, 7.44976858634351039761599644751, 8.637889565453380379885134783690, 9.294382973529502197494968858081, 9.952286091076822210115512556757, 10.94177562972859640441544698186

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