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Properties

Label 2-637-91.74-c1-0-42
Degree $2$
Conductor $637$
Sign $0.328 - 0.944i$
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  + 0.231·2-s + (−1.66 − 2.87i)3-s − 1.94·4-s + (−1.11 − 1.93i)5-s + (−0.384 − 0.665i)6-s − 0.913·8-s + (−4.01 + 6.96i)9-s + (−0.258 − 0.447i)10-s + (−1.66 − 2.87i)11-s + (3.23 + 5.60i)12-s + (−3.40 − 1.19i)13-s + (−3.70 + 6.41i)15-s + 3.68·16-s + 1.37·17-s + (−0.929 + 1.61i)18-s + (1.61 − 2.80i)19-s + ⋯
L(s)  = 1  + 0.163·2-s + (−0.959 − 1.66i)3-s − 0.973·4-s + (−0.498 − 0.864i)5-s + (−0.156 − 0.271i)6-s − 0.322·8-s + (−1.33 + 2.32i)9-s + (−0.0816 − 0.141i)10-s + (−0.500 − 0.867i)11-s + (0.933 + 1.61i)12-s + (−0.943 − 0.330i)13-s + (−0.957 + 1.65i)15-s + 0.920·16-s + 0.333·17-s + (−0.219 + 0.379i)18-s + (0.371 − 0.642i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.328 - 0.944i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.328 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.126109 + 0.0896410i\)
\(L(\frac12)\) \(\approx\) \(0.126109 + 0.0896410i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.40 + 1.19i)T \)
good2 \( 1 - 0.231T + 2T^{2} \)
3 \( 1 + (1.66 + 2.87i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.11 + 1.93i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.66 + 2.87i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 1.37T + 17T^{2} \)
19 \( 1 + (-1.61 + 2.80i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.838T + 23T^{2} \)
29 \( 1 + (-0.303 + 0.525i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.857 + 1.48i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.55T + 37T^{2} \)
41 \( 1 + (4.58 - 7.94i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.615 + 1.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.814 + 1.41i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.19 - 7.27i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 8.82T + 59T^{2} \)
61 \( 1 + (-2.73 + 4.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.09 - 8.82i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.60 - 4.51i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.98 - 3.43i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.22 + 5.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.64T + 83T^{2} \)
89 \( 1 + 9.12T + 89T^{2} \)
97 \( 1 + (7.67 + 13.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.903571289923274923687357877788, −8.627660532864009303907217465336, −8.068178119724280299131735513680, −7.33271205402128687909479868296, −6.11914656234842744851520882018, −5.28421602891361166572342331209, −4.68899178066703332277551039800, −2.87226972054584269675860941859, −1.04584439045967007931850604839, −0.11553328741105699222498847588, 3.14626964356967952620950900044, 3.97833439265206632768137301538, 4.86552681347448420102852176520, 5.41268484168231628624168043542, 6.64551248932714381739948705868, 7.79977863118772392303931209732, 9.072152276258394572586945792950, 9.816208600408797371273182378231, 10.26639436362180520545812472858, 11.07201147956892557852603260597

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