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Properties

Label 2-637-13.4-c1-0-16
Degree $2$
Conductor $637$
Sign $-0.880 - 0.473i$
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  + (1.81 + 1.04i)2-s + (−0.663 + 1.15i)3-s + (1.18 + 2.05i)4-s + 3.48i·5-s + (−2.40 + 1.38i)6-s + 0.780i·8-s + (0.618 + 1.07i)9-s + (−3.64 + 6.31i)10-s + (−0.817 − 0.472i)11-s − 3.15·12-s + (3.60 − 0.0632i)13-s + (−4.01 − 2.31i)15-s + (1.55 − 2.69i)16-s + (−2.12 − 3.68i)17-s + 2.58i·18-s + (−2.95 + 1.70i)19-s + ⋯
L(s)  = 1  + (1.28 + 0.739i)2-s + (−0.383 + 0.663i)3-s + (0.593 + 1.02i)4-s + 1.56i·5-s + (−0.981 + 0.566i)6-s + 0.275i·8-s + (0.206 + 0.356i)9-s + (−1.15 + 1.99i)10-s + (−0.246 − 0.142i)11-s − 0.909·12-s + (0.999 − 0.0175i)13-s + (−1.03 − 0.598i)15-s + (0.389 − 0.674i)16-s + (−0.516 − 0.894i)17-s + 0.609i·18-s + (−0.678 + 0.391i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.880 - 0.473i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.880 - 0.473i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.621957 + 2.47069i\)
\(L(\frac12)\) \(\approx\) \(0.621957 + 2.47069i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.60 + 0.0632i)T \)
good2 \( 1 + (-1.81 - 1.04i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.663 - 1.15i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 3.48iT - 5T^{2} \)
11 \( 1 + (0.817 + 0.472i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.12 + 3.68i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.95 - 1.70i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.72 + 4.72i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.54 - 7.86i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.15iT - 31T^{2} \)
37 \( 1 + (-3.81 - 2.19i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-8.09 - 4.67i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.84 - 6.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.96iT - 47T^{2} \)
53 \( 1 - 7.50T + 53T^{2} \)
59 \( 1 + (-9.25 + 5.34i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.44 + 5.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.42 + 3.70i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (11.3 - 6.56i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 - 12.2T + 79T^{2} \)
83 \( 1 + 1.54iT - 83T^{2} \)
89 \( 1 + (-3.52 - 2.03i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.32 + 0.765i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.87373346247090610596278546702, −10.49212475848250840606655082680, −9.341912805615996145389264372835, −7.909066499951342055733048104173, −6.98908342598577462665464496720, −6.37611862608379884406857684037, −5.51027283447783123133721229642, −4.51894130129799405220033508150, −3.65113071283398415681766465134, −2.64228896679447810929026629691, 1.04787100351110692494117694745, 2.12052059623765485909129700643, 3.89089071971230495225143561335, 4.38495278203581897050622445725, 5.65617287684519602936237536031, 6.02797472572192446390981768138, 7.47850645091637976357344260753, 8.579883863108950985214983985904, 9.288268585337115610437087869714, 10.64989698945663620845764665878

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