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Properties

Label 2-637-91.74-c1-0-5
Degree $2$
Conductor $637$
Sign $-0.835 - 0.548i$
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  − 2.33·2-s + (1.15 + 1.99i)3-s + 3.43·4-s + (1.68 + 2.91i)5-s + (−2.69 − 4.66i)6-s − 3.34·8-s + (−1.16 + 2.01i)9-s + (−3.92 − 6.80i)10-s + (−1.16 − 2.01i)11-s + (3.96 + 6.87i)12-s + (−0.408 + 3.58i)13-s + (−3.89 + 6.74i)15-s + 0.933·16-s + 5.45·17-s + (2.71 − 4.70i)18-s + (−3.58 + 6.20i)19-s + ⋯
L(s)  = 1  − 1.64·2-s + (0.666 + 1.15i)3-s + 1.71·4-s + (0.753 + 1.30i)5-s + (−1.09 − 1.90i)6-s − 1.18·8-s + (−0.388 + 0.673i)9-s + (−1.24 − 2.15i)10-s + (−0.351 − 0.608i)11-s + (1.14 + 1.98i)12-s + (−0.113 + 0.993i)13-s + (−1.00 + 1.74i)15-s + 0.233·16-s + 1.32·17-s + (0.640 − 1.10i)18-s + (−0.822 + 1.42i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.835 - 0.548i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.835 - 0.548i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.252451 + 0.844441i\)
\(L(\frac12)\) \(\approx\) \(0.252451 + 0.844441i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.408 - 3.58i)T \)
good2 \( 1 + 2.33T + 2T^{2} \)
3 \( 1 + (-1.15 - 1.99i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.68 - 2.91i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.16 + 2.01i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 5.45T + 17T^{2} \)
19 \( 1 + (3.58 - 6.20i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.45T + 23T^{2} \)
29 \( 1 + (-4.22 + 7.31i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.52 - 2.64i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.05T + 37T^{2} \)
41 \( 1 + (0.468 - 0.812i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.04 - 3.54i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.73 + 2.99i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.17 + 2.02i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 7.24T + 59T^{2} \)
61 \( 1 + (-3.19 + 5.53i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.30 + 3.99i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.79 - 6.57i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.03 - 1.79i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.79 - 6.57i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.89T + 83T^{2} \)
89 \( 1 + 13.1T + 89T^{2} \)
97 \( 1 + (1.77 + 3.08i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.40674757555282032997452833836, −9.903321990126708379427809311791, −9.588823870713672217835834768173, −8.397817216332858537120344564562, −7.88309219720193776464377792566, −6.64531929480115688973155720272, −5.89271212821711254313855585216, −4.07960432695075579383628748896, −2.96319370209523764930603837497, −1.93736358616088475201393040636, 0.76423655163299994550154811930, 1.72888373508026676429052858210, 2.65695754868378917317304619241, 4.86142367958467943862486740856, 6.04599615811058381188619099883, 7.18909228159948037538798500749, 7.85889507484750269714745716379, 8.481226904589051868692541887940, 9.141913273120150323645932037999, 9.959248288946991702269254996789

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