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Properties

Label 2-637-13.10-c1-0-6
Degree $2$
Conductor $637$
Sign $-0.964 + 0.265i$
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  + (−0.395 + 0.228i)2-s + (1.39 + 2.41i)3-s + (−0.895 + 1.55i)4-s + 0.456i·5-s + (−1.10 − 0.637i)6-s − 1.73i·8-s + (−2.39 + 4.14i)9-s + (−0.104 − 0.180i)10-s + (−3.39 + 1.96i)11-s − 5·12-s + (−3.5 − 0.866i)13-s + (−1.10 + 0.637i)15-s + (−1.39 − 2.41i)16-s + (1.5 − 2.59i)17-s − 2.18i·18-s + (−1.18 − 0.685i)19-s + ⋯
L(s)  = 1  + (−0.279 + 0.161i)2-s + (0.805 + 1.39i)3-s + (−0.447 + 0.775i)4-s + 0.204i·5-s + (−0.450 − 0.260i)6-s − 0.612i·8-s + (−0.798 + 1.38i)9-s + (−0.0330 − 0.0571i)10-s + (−1.02 + 0.591i)11-s − 1.44·12-s + (−0.970 − 0.240i)13-s + (−0.285 + 0.164i)15-s + (−0.348 − 0.604i)16-s + (0.363 − 0.630i)17-s − 0.515i·18-s + (−0.272 − 0.157i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.964 + 0.265i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.964 + 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.139667 - 1.03522i\)
\(L(\frac12)\) \(\approx\) \(0.139667 - 1.03522i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.5 + 0.866i)T \)
good2 \( 1 + (0.395 - 0.228i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.39 - 2.41i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 0.456iT - 5T^{2} \)
11 \( 1 + (3.39 - 1.96i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.18 + 0.685i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.791 - 1.37i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.39 - 5.88i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.66iT - 31T^{2} \)
37 \( 1 + (-6 + 3.46i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.79 - 3.92i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.68 - 8.11i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 9.57iT - 47T^{2} \)
53 \( 1 - 6.16T + 53T^{2} \)
59 \( 1 + (10.6 + 6.15i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.37 - 12.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.87 - 2.23i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.79 - 2.18i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 3.46iT - 73T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
83 \( 1 + 7.02iT - 83T^{2} \)
89 \( 1 + (-13.9 + 8.07i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.31 - 3.64i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.51282413827887361165080505597, −10.13464250534185857531211473455, −9.258714560175922309285970252515, −8.627512684767130511465316451076, −7.74877425168365594260237677889, −6.98189212705122811679862003759, −4.96206817363485237646196803907, −4.74073964159259720591452148693, −3.29498762492748357425125107314, −2.77811873909924684767672166968, 0.55432840540967627536145525108, 1.92137794749078930091982327511, 2.84146579155933846267424105971, 4.56667528476319610584813868021, 5.72595265434186874292787510418, 6.58528229148462483192431559472, 7.78203428673521880109879486901, 8.233523347732689336078091939567, 9.098994891542534246660322649804, 10.00621174066112442862171343157

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