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Properties

Label 2-637-49.39-c1-0-32
Degree $2$
Conductor $637$
Sign $0.526 + 0.849i$
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  + (0.232 + 0.0350i)2-s + (−1.30 + 0.887i)3-s + (−1.85 − 0.573i)4-s + (0.133 + 1.78i)5-s + (−0.334 + 0.160i)6-s + (−0.0133 − 2.64i)7-s + (−0.836 − 0.403i)8-s + (−0.189 + 0.481i)9-s + (−0.0314 + 0.420i)10-s + (0.106 + 0.272i)11-s + (2.92 − 0.903i)12-s + (−0.623 + 0.781i)13-s + (0.0897 − 0.616i)14-s + (−1.75 − 2.20i)15-s + (3.03 + 2.06i)16-s + (−2.56 − 2.38i)17-s + ⋯
L(s)  = 1  + (0.164 + 0.0248i)2-s + (−0.751 + 0.512i)3-s + (−0.929 − 0.286i)4-s + (0.0597 + 0.797i)5-s + (−0.136 + 0.0657i)6-s + (−0.00502 − 0.999i)7-s + (−0.295 − 0.142i)8-s + (−0.0630 + 0.160i)9-s + (−0.00995 + 0.132i)10-s + (0.0322 + 0.0820i)11-s + (0.845 − 0.260i)12-s + (−0.172 + 0.216i)13-s + (0.0239 − 0.164i)14-s + (−0.453 − 0.568i)15-s + (0.758 + 0.516i)16-s + (−0.622 − 0.577i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 + 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.526 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.526 + 0.849i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.526 + 0.849i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.589536 - 0.328128i\)
\(L(\frac12)\) \(\approx\) \(0.589536 - 0.328128i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.0133 + 2.64i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
good2 \( 1 + (-0.232 - 0.0350i)T + (1.91 + 0.589i)T^{2} \)
3 \( 1 + (1.30 - 0.887i)T + (1.09 - 2.79i)T^{2} \)
5 \( 1 + (-0.133 - 1.78i)T + (-4.94 + 0.745i)T^{2} \)
11 \( 1 + (-0.106 - 0.272i)T + (-8.06 + 7.48i)T^{2} \)
17 \( 1 + (2.56 + 2.38i)T + (1.27 + 16.9i)T^{2} \)
19 \( 1 + (1.53 + 2.66i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.55 + 3.30i)T + (1.71 - 22.9i)T^{2} \)
29 \( 1 + (-1.23 + 5.39i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-2.57 + 4.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.41 + 2.28i)T + (30.5 - 20.8i)T^{2} \)
41 \( 1 + (1.70 + 0.819i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (2.71 - 1.30i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-10.8 - 1.63i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (-6.22 - 1.91i)T + (43.7 + 29.8i)T^{2} \)
59 \( 1 + (-0.614 + 8.19i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (11.2 - 3.47i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (4.28 - 7.42i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.89 + 12.6i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (7.22 - 1.08i)T + (69.7 - 21.5i)T^{2} \)
79 \( 1 + (2.67 + 4.64i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.24 + 11.5i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-0.677 + 1.72i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + 15.8T + 97T^{2} \)
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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

−10.54464511004730497279981646046, −9.802270809679046036544005378204, −8.900865024463641448061914007324, −7.69678870344432877222729944169, −6.70879871180894044139380714500, −5.85477524831329443073060245344, −4.58885437379613573549352267262, −4.33313411341194170086309170459, −2.73912977194102638862099104234, −0.45411441930331718293613047855, 1.22425064360962829223077753867, 3.06372256678235238310246499175, 4.42404410141430596436402904479, 5.35162254851452789574199904761, 5.91859136688945076093128140217, 7.08559573158821456317797601082, 8.466292974385188492093004574639, 8.774131781560657599052360982219, 9.658591832929514040989044819384, 10.87600702371740527442162634597

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