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Properties

Label 2-630-15.8-c1-0-9
Degree $2$
Conductor $630$
Sign $0.360 + 0.932i$
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 + (2.23 + 0.0743i)5-s + (−0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s + (−1.52 − 1.63i)10-s − 5.26i·11-s + (−3.16 − 3.16i)13-s + 1.00·14-s − 1.00·16-s + (3.05 + 3.05i)17-s + (−0.0743 + 2.23i)20-s + (−3.72 + 3.72i)22-s + (4.32 − 4.32i)23-s + (4.98 + 0.332i)25-s + 4.46i·26-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (0.999 + 0.0332i)5-s + (−0.267 + 0.267i)7-s + (0.250 − 0.250i)8-s + (−0.483 − 0.516i)10-s − 1.58i·11-s + (−0.876 − 0.876i)13-s + 0.267·14-s − 0.250·16-s + (0.741 + 0.741i)17-s + (−0.0166 + 0.499i)20-s + (−0.793 + 0.793i)22-s + (0.901 − 0.901i)23-s + (0.997 + 0.0664i)25-s + 0.876i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.360 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.360 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.360 + 0.932i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.360 + 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06087 - 0.727570i\)
\(L(\frac12)\) \(\approx\) \(1.06087 - 0.727570i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 - 0.0743i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good11 \( 1 + 5.26iT - 11T^{2} \)
13 \( 1 + (3.16 + 3.16i)T + 13iT^{2} \)
17 \( 1 + (-3.05 - 3.05i)T + 17iT^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (-4.32 + 4.32i)T - 23iT^{2} \)
29 \( 1 - 9.96T + 29T^{2} \)
31 \( 1 - 1.26T + 31T^{2} \)
37 \( 1 + (2.93 - 2.93i)T - 37iT^{2} \)
41 \( 1 + 10.6iT - 41T^{2} \)
43 \( 1 + (-0.597 - 0.597i)T + 43iT^{2} \)
47 \( 1 + (3.42 + 3.42i)T + 47iT^{2} \)
53 \( 1 + (9.88 - 9.88i)T - 53iT^{2} \)
59 \( 1 - 3.12T + 59T^{2} \)
61 \( 1 - 3.05T + 61T^{2} \)
67 \( 1 + (-4.59 + 4.59i)T - 67iT^{2} \)
71 \( 1 + 9.23iT - 71T^{2} \)
73 \( 1 + (-10.2 - 10.2i)T + 73iT^{2} \)
79 \( 1 + 3.57iT - 79T^{2} \)
83 \( 1 + (6.21 - 6.21i)T - 83iT^{2} \)
89 \( 1 + 3.61T + 89T^{2} \)
97 \( 1 + (4.63 - 4.63i)T - 97iT^{2} \)
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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

−10.42491544930724075477188202214, −9.704118398716072866853330503898, −8.698275618178317013419999866047, −8.176506365441234368614100336666, −6.80026094228103452388699715321, −5.91768002144676876932661237047, −5.01457523714989425088478510048, −3.30385441964918558888678818341, −2.56735712387815124424376026955, −0.920430103883628314364526518907, 1.48383441661779344797512070084, 2.73347815112061311453300305453, 4.62343723239268312153534364749, 5.24471398701323641663014917790, 6.65600973436177194194011448496, 6.99853108551724290272595687407, 8.063687940509863226616997076481, 9.471635221035070876750994475326, 9.589263495757798807098579991643, 10.33322571895259133929261080025

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