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Properties

Label 2-630-35.27-c1-0-0
Degree $2$
Conductor $630$
Sign $-0.390 - 0.920i$
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  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−1.28 − 1.83i)5-s + (0.0564 + 2.64i)7-s + (0.707 − 0.707i)8-s + (−0.386 + 2.20i)10-s − 1.41·11-s + (−0.772 − 0.772i)13-s + (1.83 − 1.91i)14-s − 1.00·16-s + (0.546 − 0.546i)17-s − 5.95·19-s + (1.83 − 1.28i)20-s + (1.00 + 1.00i)22-s + (−5.23 + 5.23i)23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.574 − 0.818i)5-s + (0.0213 + 0.999i)7-s + (0.250 − 0.250i)8-s + (−0.122 + 0.696i)10-s − 0.426·11-s + (−0.214 − 0.214i)13-s + (0.489 − 0.510i)14-s − 0.250·16-s + (0.132 − 0.132i)17-s − 1.36·19-s + (0.409 − 0.287i)20-s + (0.213 + 0.213i)22-s + (−1.09 + 1.09i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.390 - 0.920i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.390 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.150510 + 0.227359i\)
\(L(\frac12)\) \(\approx\) \(0.150510 + 0.227359i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 + (1.28 + 1.83i)T \)
7 \( 1 + (-0.0564 - 2.64i)T \)
good11 \( 1 + 1.41T + 11T^{2} \)
13 \( 1 + (0.772 + 0.772i)T + 13iT^{2} \)
17 \( 1 + (-0.546 + 0.546i)T - 17iT^{2} \)
19 \( 1 + 5.95T + 19T^{2} \)
23 \( 1 + (5.23 - 5.23i)T - 23iT^{2} \)
29 \( 1 + 0.992iT - 29T^{2} \)
31 \( 1 - 2.85iT - 31T^{2} \)
37 \( 1 + (-5.70 - 5.70i)T + 37iT^{2} \)
41 \( 1 + 2.18iT - 41T^{2} \)
43 \( 1 + (2 - 2i)T - 43iT^{2} \)
47 \( 1 + (7.32 - 7.32i)T - 47iT^{2} \)
53 \( 1 + (2.40 - 2.40i)T - 53iT^{2} \)
59 \( 1 + 12.0T + 59T^{2} \)
61 \( 1 + 3.63iT - 61T^{2} \)
67 \( 1 + (4 + 4i)T + 67iT^{2} \)
71 \( 1 - 8.06T + 71T^{2} \)
73 \( 1 + (-11.3 - 11.3i)T + 73iT^{2} \)
79 \( 1 + 5.40iT - 79T^{2} \)
83 \( 1 + (-8.79 - 8.79i)T + 83iT^{2} \)
89 \( 1 + 11.3T + 89T^{2} \)
97 \( 1 + (-7.76 + 7.76i)T - 97iT^{2} \)
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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

−10.98005117402729819284807107674, −9.829312818108683686019850637180, −9.194323819126184959393208241405, −8.196386720702979172708916081702, −7.87979751344850867143715490754, −6.37662740009318528736699856952, −5.27279476231311554401035925638, −4.29132894147206517828919743012, −3.01169719538813656573884155471, −1.72659815414085056347874032648, 0.16829979734092600541731888593, 2.24920965438562494954385432889, 3.77850253168192721944252783782, 4.65539541673491147240405114671, 6.20113323358228526841148377264, 6.79168369847891939597147715664, 7.78093824014653872930592105819, 8.235162159303425422364067877619, 9.533860482781788103342601515071, 10.51330095537759631609151679493

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