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Properties

Label 2-630-315.194-c1-0-23
Degree $2$
Conductor $630$
Sign $0.739 - 0.672i$
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 + (−1.48 + 0.894i)3-s + 4-s + (1.24 + 1.86i)5-s + (−1.48 + 0.894i)6-s + (2.20 − 1.46i)7-s + 8-s + (1.40 − 2.65i)9-s + (1.24 + 1.86i)10-s + (0.646 + 0.372i)11-s + (−1.48 + 0.894i)12-s + (1.67 − 2.89i)13-s + (2.20 − 1.46i)14-s + (−3.50 − 1.64i)15-s + 16-s + (−4.20 + 2.42i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.856 + 0.516i)3-s + 0.5·4-s + (0.555 + 0.831i)5-s + (−0.605 + 0.365i)6-s + (0.832 − 0.554i)7-s + 0.353·8-s + (0.466 − 0.884i)9-s + (0.392 + 0.588i)10-s + (0.194 + 0.112i)11-s + (−0.428 + 0.258i)12-s + (0.463 − 0.802i)13-s + (0.588 − 0.391i)14-s + (−0.904 − 0.425i)15-s + 0.250·16-s + (−1.01 + 0.588i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.739 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.739 - 0.672i$
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: \((2,\ 630,\ (\ :1/2),\ 0.739 - 0.672i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98961 + 0.769353i\)
\(L(\frac12)\) \(\approx\) \(1.98961 + 0.769353i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.48 - 0.894i)T \)
5 \( 1 + (-1.24 - 1.86i)T \)
7 \( 1 + (-2.20 + 1.46i)T \)
good11 \( 1 + (-0.646 - 0.372i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.67 + 2.89i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.20 - 2.42i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.50 - 3.75i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.933 + 1.61i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.644 - 0.371i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.00iT - 31T^{2} \)
37 \( 1 + (5.78 + 3.33i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.849 + 1.47i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.64 + 4.41i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 11.6iT - 47T^{2} \)
53 \( 1 + (2.94 + 5.09i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 4.66T + 59T^{2} \)
61 \( 1 + 1.71iT - 61T^{2} \)
67 \( 1 - 4.53iT - 67T^{2} \)
71 \( 1 + 4.17iT - 71T^{2} \)
73 \( 1 + (5.10 + 8.84i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 6.21T + 79T^{2} \)
83 \( 1 + (2.26 - 1.30i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.58 + 13.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.32 - 4.01i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.64827159725333006907661148373, −10.37944274664346192968370031170, −9.174910389971236278297339154836, −7.76463412017517643682227961218, −6.89438375281546563004304652991, −5.97602966195332291921140913043, −5.29193143326389256183385965392, −4.20218429784113522547464558478, −3.26448567525974530868194860285, −1.55241008826991086691232284832, 1.28525791221713083435081951024, 2.36337579052772002400886849903, 4.33308625181246326207277916756, 5.07752995277257203207652511304, 5.75309054834104578077005765578, 6.68846879869719797804674000271, 7.64832234732812824715453876091, 8.776188635352205914803782238068, 9.585266143977494516712707078010, 10.91699144360711602376678822961

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