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Properties

Label 2-637-91.54-c1-0-5
Degree $2$
Conductor $637$
Sign $0.755 - 0.655i$
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  + (−1.42 + 1.42i)2-s + (−1.25 − 0.721i)3-s − 2.06i·4-s + (−3.16 + 0.849i)5-s + (2.81 − 0.753i)6-s + (0.0872 + 0.0872i)8-s + (−0.457 − 0.793i)9-s + (3.30 − 5.72i)10-s + (−5.74 + 1.53i)11-s + (−1.48 + 2.57i)12-s + (−2.81 + 2.25i)13-s + (4.57 + 1.22i)15-s + 3.87·16-s + 0.628·17-s + (1.78 + 0.477i)18-s + (0.191 − 0.712i)19-s + ⋯
L(s)  = 1  + (−1.00 + 1.00i)2-s + (−0.721 − 0.416i)3-s − 1.03i·4-s + (−1.41 + 0.379i)5-s + (1.14 − 0.307i)6-s + (0.0308 + 0.0308i)8-s + (−0.152 − 0.264i)9-s + (1.04 − 1.81i)10-s + (−1.73 + 0.463i)11-s + (−0.429 + 0.743i)12-s + (−0.779 + 0.626i)13-s + (1.18 + 0.316i)15-s + 0.968·16-s + 0.152·17-s + (0.420 + 0.112i)18-s + (0.0438 − 0.163i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.755 - 0.655i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.755 - 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.189162 + 0.0706245i\)
\(L(\frac12)\) \(\approx\) \(0.189162 + 0.0706245i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.81 - 2.25i)T \)
good2 \( 1 + (1.42 - 1.42i)T - 2iT^{2} \)
3 \( 1 + (1.25 + 0.721i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (3.16 - 0.849i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (5.74 - 1.53i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 - 0.628T + 17T^{2} \)
19 \( 1 + (-0.191 + 0.712i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 4.54iT - 23T^{2} \)
29 \( 1 + (1.33 + 2.31i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.285 + 1.06i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.79 - 1.79i)T + 37iT^{2} \)
41 \( 1 + (-0.746 + 2.78i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-3.49 - 2.01i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.90 - 7.10i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.89 - 6.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.673 - 0.673i)T - 59iT^{2} \)
61 \( 1 + (-0.943 + 0.544i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.39 + 5.21i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.590 - 2.20i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-5.94 - 1.59i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (6.08 - 10.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.59 + 3.59i)T + 83iT^{2} \)
89 \( 1 + (-2.88 + 2.88i)T - 89iT^{2} \)
97 \( 1 + (8.41 - 2.25i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.62782155203376632082655341807, −9.704555337564836525736283986966, −8.641402224101788300846971090782, −7.72745120641540955828683264903, −7.38137078073969733509751906908, −6.57410490983111790505454304669, −5.54272959296340931234818760696, −4.36715507676329571327961264407, −2.83147244623240863875608108544, −0.39219856590325524806086196161, 0.48814949855025512373161254245, 2.52499493628136366170979592982, 3.59037759106295668551829291181, 5.00073751867944331045807498552, 5.57649714634401376098965221471, 7.57918093777895150047336713510, 7.939821766570361247648893952118, 8.773598468518559652104469789082, 9.990022735977825834082794077611, 10.53433228942265357200105118457

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