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Properties

Label 2-630-315.212-c1-0-37
Degree $2$
Conductor $630$
Sign $-0.577 + 0.816i$
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 + (−0.821 − 1.52i)3-s − 1.00i·4-s + (2.22 + 0.221i)5-s + (−1.65 − 0.497i)6-s + (1.18 − 2.36i)7-s + (−0.707 − 0.707i)8-s + (−1.65 + 2.50i)9-s + (1.72 − 1.41i)10-s + (2.40 + 1.38i)11-s + (−1.52 + 0.821i)12-s + (−1.49 − 5.58i)13-s + (−0.835 − 2.51i)14-s + (−1.49 − 3.57i)15-s − 1.00·16-s + (−2.92 − 0.783i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.474 − 0.880i)3-s − 0.500i·4-s + (0.995 + 0.0988i)5-s + (−0.677 − 0.203i)6-s + (0.447 − 0.894i)7-s + (−0.250 − 0.250i)8-s + (−0.550 + 0.835i)9-s + (0.546 − 0.448i)10-s + (0.724 + 0.418i)11-s + (−0.440 + 0.237i)12-s + (−0.414 − 1.54i)13-s + (−0.223 − 0.670i)14-s + (−0.384 − 0.922i)15-s − 0.250·16-s + (−0.709 − 0.190i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.902080 - 1.74343i\)
\(L(\frac12)\) \(\approx\) \(0.902080 - 1.74343i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.821 + 1.52i)T \)
5 \( 1 + (-2.22 - 0.221i)T \)
7 \( 1 + (-1.18 + 2.36i)T \)
good11 \( 1 + (-2.40 - 1.38i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.49 + 5.58i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (2.92 + 0.783i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-5.38 - 3.11i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.0280 - 0.104i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.103 + 0.178i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.08T + 31T^{2} \)
37 \( 1 + (5.62 - 1.50i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.812 - 0.468i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (10.6 + 2.85i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (3.35 - 3.35i)T - 47iT^{2} \)
53 \( 1 + (-0.593 + 2.21i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 - 14.4T + 59T^{2} \)
61 \( 1 - 2.11T + 61T^{2} \)
67 \( 1 + (-10.7 - 10.7i)T + 67iT^{2} \)
71 \( 1 - 12.6iT - 71T^{2} \)
73 \( 1 + (-1.62 - 0.436i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 - 2.67iT - 79T^{2} \)
83 \( 1 + (-1.61 + 6.03i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-5.61 + 9.72i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.285 - 1.06i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.32606472578383119227461708103, −9.866327033848275358740237825390, −8.482095899137594516569254585545, −7.35685485573130287223537498138, −6.70048122575639447127977480234, −5.55084729027292731234624874039, −5.01001983783963307978064358426, −3.45150646748423516138768594822, −2.09840288131142194968427392754, −1.04865469029640062208024233868, 2.03061097365401121001909577328, 3.49529080643401565390774663491, 4.77223724356809536999915354569, 5.28259370226467117626260467750, 6.27872213288762602024908406993, 6.91503574701028729625341878618, 8.649874686150052579882283373315, 9.138491479688993538005366015432, 9.789108067549472872712265094375, 11.16229970512238690815939451032

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