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Properties

Label 2-630-315.23-c1-0-13
Degree $2$
Conductor $630$
Sign $-0.921 - 0.388i$
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 + (0.300 + 1.70i)3-s + 1.00i·4-s + (2.15 + 0.595i)5-s + (−0.993 + 1.41i)6-s + (−2.63 + 0.260i)7-s + (−0.707 + 0.707i)8-s + (−2.81 + 1.02i)9-s + (1.10 + 1.94i)10-s + (−0.920 − 0.531i)11-s + (−1.70 + 0.300i)12-s + (−5.82 + 1.56i)13-s + (−2.04 − 1.67i)14-s + (−0.367 + 3.85i)15-s − 1.00·16-s + (−0.372 + 1.39i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.173 + 0.984i)3-s + 0.500i·4-s + (0.963 + 0.266i)5-s + (−0.405 + 0.579i)6-s + (−0.995 + 0.0985i)7-s + (−0.250 + 0.250i)8-s + (−0.939 + 0.341i)9-s + (0.348 + 0.615i)10-s + (−0.277 − 0.160i)11-s + (−0.492 + 0.0867i)12-s + (−1.61 + 0.433i)13-s + (−0.546 − 0.448i)14-s + (−0.0949 + 0.995i)15-s − 0.250·16-s + (−0.0903 + 0.337i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 - 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 - 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.921 - 0.388i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.921 - 0.388i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.355065 + 1.75455i\)
\(L(\frac12)\) \(\approx\) \(0.355065 + 1.75455i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.300 - 1.70i)T \)
5 \( 1 + (-2.15 - 0.595i)T \)
7 \( 1 + (2.63 - 0.260i)T \)
good11 \( 1 + (0.920 + 0.531i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.82 - 1.56i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.372 - 1.39i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.99 - 1.14i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-8.59 - 2.30i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.36 - 2.36i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.68T + 31T^{2} \)
37 \( 1 + (-0.681 - 2.54i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.580 + 0.334i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.513 - 1.91i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (1.10 + 1.10i)T + 47iT^{2} \)
53 \( 1 + (6.65 + 1.78i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 - 8.10T + 59T^{2} \)
61 \( 1 - 1.52T + 61T^{2} \)
67 \( 1 + (-10.4 + 10.4i)T - 67iT^{2} \)
71 \( 1 + 1.27iT - 71T^{2} \)
73 \( 1 + (2.63 - 9.82i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + 14.4iT - 79T^{2} \)
83 \( 1 + (9.19 + 2.46i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (3.87 - 6.71i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.457 - 0.122i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.78495039009339520102116916107, −9.821828980724761297614270903738, −9.518512140280952732618154545874, −8.501193351429460877805924064166, −7.19036353923422902040415633379, −6.39816464840504869128476148676, −5.36959538204395464786468585379, −4.74092180922701224768742275455, −3.28686535524482392449683618146, −2.59472116112964050248136815154, 0.792171467310280591294564694889, 2.47798151280217767304562264743, 2.92383255138575004251517345786, 4.79867334029258552334409234017, 5.60913925790227231238614371798, 6.65807037451579600603932115271, 7.24410908176347099918473552241, 8.603431455139717388329091776172, 9.598046996052931944749334890272, 10.04111527266610691460495532011

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