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Properties

Label 2-630-315.194-c1-0-26
Degree $2$
Conductor $630$
Sign $0.644 + 0.764i$
Analytic cond. $5.03057$
Root an. cond. $2.24289$
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  − 2-s + (1.70 − 0.322i)3-s + 4-s + (−2.15 − 0.613i)5-s + (−1.70 + 0.322i)6-s + (1.39 + 2.24i)7-s − 8-s + (2.79 − 1.09i)9-s + (2.15 + 0.613i)10-s + (−4.57 − 2.63i)11-s + (1.70 − 0.322i)12-s + (2.80 − 4.86i)13-s + (−1.39 − 2.24i)14-s + (−3.85 − 0.350i)15-s + 16-s + (4.30 − 2.48i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.982 − 0.186i)3-s + 0.5·4-s + (−0.961 − 0.274i)5-s + (−0.694 + 0.131i)6-s + (0.527 + 0.849i)7-s − 0.353·8-s + (0.930 − 0.366i)9-s + (0.679 + 0.194i)10-s + (−1.37 − 0.795i)11-s + (0.491 − 0.0931i)12-s + (0.778 − 1.34i)13-s + (−0.373 − 0.600i)14-s + (−0.995 − 0.0904i)15-s + 0.250·16-s + (1.04 − 0.603i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.644 + 0.764i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.644 + 0.764i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.644 + 0.764i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20065 - 0.558032i\)
\(L(\frac12)\) \(\approx\) \(1.20065 - 0.558032i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-1.70 + 0.322i)T \)
5 \( 1 + (2.15 + 0.613i)T \)
7 \( 1 + (-1.39 - 2.24i)T \)
good11 \( 1 + (4.57 + 2.63i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.80 + 4.86i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-4.30 + 2.48i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0258 + 0.0149i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.13 - 3.70i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.48 + 3.74i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.77iT - 31T^{2} \)
37 \( 1 + (-1.82 - 1.05i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.423 + 0.733i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.31 - 4.22i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.59iT - 47T^{2} \)
53 \( 1 + (-0.914 - 1.58i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 2.43T + 59T^{2} \)
61 \( 1 - 4.23iT - 61T^{2} \)
67 \( 1 + 14.2iT - 67T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + (2.23 + 3.87i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 4.22T + 79T^{2} \)
83 \( 1 + (9.84 - 5.68i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.09 - 8.83i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.01 - 10.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.38527910786260486868770032631, −9.411165360519034543915139520592, −8.421895408758483829507932192506, −7.986694728025182393858899282250, −7.62315577891522757982417302486, −5.99197763663203158352386908237, −4.98073024572547406605256279024, −3.35159548470354655943343674803, −2.71047892735835603917362900382, −0.919413773679325048257183300603, 1.49304290132056574720824230241, 2.92350731755832059571208744660, 3.99732423674040964499910534535, 4.89467201648035721756757082154, 6.91791085823385457633524874689, 7.26629000067612817388565612591, 8.415083315697234086156751543894, 8.498501321088479554699501565343, 10.10357623691382825628121825329, 10.38370653970068384340124893274

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