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Properties

Label 2-637-91.51-c1-0-28
Degree $2$
Conductor $637$
Sign $-0.962 + 0.271i$
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  + (−1.01 − 0.585i)2-s + (−0.269 − 0.466i)3-s + (−0.315 − 0.546i)4-s + (0.399 + 0.230i)5-s + 0.630i·6-s + 3.07i·8-s + (1.35 − 2.34i)9-s + (−0.269 − 0.466i)10-s + (0.718 − 0.414i)11-s + (−0.170 + 0.294i)12-s + (−2.87 − 2.17i)13-s − 0.248i·15-s + (1.17 − 2.02i)16-s + (−1.43 − 2.49i)17-s + (−2.74 + 1.58i)18-s + (3.74 + 2.16i)19-s + ⋯
L(s)  = 1  + (−0.716 − 0.413i)2-s + (−0.155 − 0.269i)3-s + (−0.157 − 0.273i)4-s + (0.178 + 0.103i)5-s + 0.257i·6-s + 1.08i·8-s + (0.451 − 0.782i)9-s + (−0.0852 − 0.147i)10-s + (0.216 − 0.125i)11-s + (−0.0490 + 0.0850i)12-s + (−0.798 − 0.601i)13-s − 0.0641i·15-s + (0.292 − 0.506i)16-s + (−0.349 − 0.604i)17-s + (−0.647 + 0.373i)18-s + (0.859 + 0.496i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.962 + 0.271i$
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: \((2,\ 637,\ (\ :1/2),\ -0.962 + 0.271i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0855201 - 0.618947i\)
\(L(\frac12)\) \(\approx\) \(0.0855201 - 0.618947i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.87 + 2.17i)T \)
good2 \( 1 + (1.01 + 0.585i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.269 + 0.466i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.399 - 0.230i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.718 + 0.414i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.43 + 2.49i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.74 - 2.16i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.52 + 4.37i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.261T + 29T^{2} \)
31 \( 1 + (5.88 - 3.40i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (8.23 + 4.75i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.68iT - 41T^{2} \)
43 \( 1 - 0.418T + 43T^{2} \)
47 \( 1 + (8.00 + 4.62i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.815 + 1.41i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.41 - 1.39i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.63 - 6.28i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.39 + 2.53i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.7iT - 71T^{2} \)
73 \( 1 + (-0.306 + 0.176i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.40 - 2.43i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.3iT - 83T^{2} \)
89 \( 1 + (4.70 + 2.71i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.6iT - 97T^{2} \)
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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

−10.07765635734109679177833191021, −9.467758591261903442245975691077, −8.680173793911866351796499355464, −7.61948712053646935309387841286, −6.69879053939664729837265416294, −5.64324919499359942685885019628, −4.70384676921721884707547567096, −3.17962772386370803737750629418, −1.81145465402420779534831176708, −0.44729157853139936097179006898, 1.74002954589489246450885825079, 3.47793242926042702291639299403, 4.56069362953612046552111139715, 5.50044514538294521679650515110, 6.95708640375813501028442544302, 7.39747200314462616998503824711, 8.376672862841769474483903917321, 9.430594464019065867192787632743, 9.693394836878277324945204006846, 10.82513315397161700210269154615

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