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Properties

Label 2-637-91.59-c1-0-38
Degree $2$
Conductor $637$
Sign $0.999 - 0.0432i$
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  + (1.28 + 1.28i)2-s + (2.65 − 1.53i)3-s + 1.30i·4-s + (−1.07 − 0.288i)5-s + (5.38 + 1.44i)6-s + (0.899 − 0.899i)8-s + (3.21 − 5.56i)9-s + (−1.01 − 1.75i)10-s + (0.124 + 0.0332i)11-s + (1.99 + 3.45i)12-s + (−3.59 + 0.291i)13-s + (−3.30 + 0.884i)15-s + 4.90·16-s + 0.273·17-s + (11.2 − 3.02i)18-s + (1.02 + 3.83i)19-s + ⋯
L(s)  = 1  + (0.908 + 0.908i)2-s + (1.53 − 0.886i)3-s + 0.650i·4-s + (−0.480 − 0.128i)5-s + (2.19 + 0.589i)6-s + (0.317 − 0.317i)8-s + (1.07 − 1.85i)9-s + (−0.319 − 0.553i)10-s + (0.0374 + 0.0100i)11-s + (0.576 + 0.997i)12-s + (−0.996 + 0.0807i)13-s + (−0.852 + 0.228i)15-s + 1.22·16-s + 0.0663·17-s + (2.65 − 0.711i)18-s + (0.235 + 0.879i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0432i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0432i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.999 - 0.0432i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.999 - 0.0432i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.56926 + 0.0772760i\)
\(L(\frac12)\) \(\approx\) \(3.56926 + 0.0772760i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.59 - 0.291i)T \)
good2 \( 1 + (-1.28 - 1.28i)T + 2iT^{2} \)
3 \( 1 + (-2.65 + 1.53i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.07 + 0.288i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.124 - 0.0332i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 - 0.273T + 17T^{2} \)
19 \( 1 + (-1.02 - 3.83i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 0.491iT - 23T^{2} \)
29 \( 1 + (4.62 - 8.00i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.85 - 6.92i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.82 - 2.82i)T - 37iT^{2} \)
41 \( 1 + (-2.31 - 8.63i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-8.44 + 4.87i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.51 + 5.66i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.467 + 0.809i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.86 + 4.86i)T + 59iT^{2} \)
61 \( 1 + (6.74 + 3.89i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.316 - 1.18i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.793 - 2.96i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.118 - 0.0318i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.72 + 9.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.07 + 8.07i)T - 83iT^{2} \)
89 \( 1 + (2.99 + 2.99i)T + 89iT^{2} \)
97 \( 1 + (2.39 + 0.641i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.39196028207788681850775355036, −9.432201986513988598100909147985, −8.480844954977569467968989425082, −7.64759242770104196653605823871, −7.23355973996004037591496021073, −6.30966338126448223700143545603, −5.05117445425975898066579888636, −3.94090731165345067736006973484, −3.07733611755958460729481178730, −1.60645932166581575986381109138, 2.24425296373830769223549522906, 2.86442533360242888952055206854, 3.97275071603986584130652186667, 4.36057054191802402849510411771, 5.53166459247862916937231361037, 7.54862671715638158047511113634, 7.83040298499949228610543682184, 9.145980103358099032993303854987, 9.691365561785637135733597178333, 10.65321570118351170239194773805

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