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Properties

Label 2-630-315.194-c1-0-44
Degree $2$
Conductor $630$
Sign $0.0734 + 0.997i$
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 + (0.193 − 1.72i)3-s + 4-s + (2.07 + 0.842i)5-s + (0.193 − 1.72i)6-s + (−1.57 − 2.12i)7-s + 8-s + (−2.92 − 0.667i)9-s + (2.07 + 0.842i)10-s + (−4.72 − 2.73i)11-s + (0.193 − 1.72i)12-s + (2.33 − 4.04i)13-s + (−1.57 − 2.12i)14-s + (1.85 − 3.40i)15-s + 16-s + (−0.153 + 0.0885i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.111 − 0.993i)3-s + 0.5·4-s + (0.926 + 0.376i)5-s + (0.0791 − 0.702i)6-s + (−0.594 − 0.804i)7-s + 0.353·8-s + (−0.974 − 0.222i)9-s + (0.655 + 0.266i)10-s + (−1.42 − 0.823i)11-s + (0.0559 − 0.496i)12-s + (0.648 − 1.12i)13-s + (−0.420 − 0.568i)14-s + (0.477 − 0.878i)15-s + 0.250·16-s + (−0.0372 + 0.0214i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0734 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0734 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.0734 + 0.997i$
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: \((2,\ 630,\ (\ :1/2),\ 0.0734 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74667 - 1.62275i\)
\(L(\frac12)\) \(\approx\) \(1.74667 - 1.62275i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.193 + 1.72i)T \)
5 \( 1 + (-2.07 - 0.842i)T \)
7 \( 1 + (1.57 + 2.12i)T \)
good11 \( 1 + (4.72 + 2.73i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.33 + 4.04i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.153 - 0.0885i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.69 - 2.13i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.10 - 1.91i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-8.66 + 5.00i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.76iT - 31T^{2} \)
37 \( 1 + (-5.07 - 2.93i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.20 - 9.01i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.58 - 1.49i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.22iT - 47T^{2} \)
53 \( 1 + (-5.40 - 9.36i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 9.73T + 59T^{2} \)
61 \( 1 + 9.74iT - 61T^{2} \)
67 \( 1 + 11.4iT - 67T^{2} \)
71 \( 1 - 12.4iT - 71T^{2} \)
73 \( 1 + (0.142 + 0.246i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 5.83T + 79T^{2} \)
83 \( 1 + (5.44 - 3.14i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.64 - 4.57i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.26 - 2.19i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.51562911329654925446428959741, −9.786147142433101942258264217996, −8.258892974893396357606023844650, −7.68490510185483966649129763879, −6.56166099164831140670189581005, −5.97581411887904960045497311977, −5.15279999816023545796386007163, −3.24159998928801197097125729101, −2.79724096391663233444096461830, −1.07561771353199848087099645335, 2.22867365728370627381553661354, 3.06357719040924271881015821989, 4.52440737323284850998079616104, 5.18870983588666592276789468319, 5.95883477763734631064316026774, 6.98631461843367075709100908954, 8.488575942315609936023334076044, 9.179946123881386077776206963803, 10.00085848887744131623410045767, 10.63156409861418632335706714220

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