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Properties

Label 2-630-315.23-c1-0-41
Degree $2$
Conductor $630$
Sign $-0.782 + 0.623i$
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 + (−0.717 + 1.57i)3-s + 1.00i·4-s + (2.15 − 0.609i)5-s + (1.62 − 0.607i)6-s + (−0.673 − 2.55i)7-s + (0.707 − 0.707i)8-s + (−1.97 − 2.26i)9-s + (−1.95 − 1.09i)10-s + (−4.13 − 2.38i)11-s + (−1.57 − 0.717i)12-s + (−1.30 + 0.348i)13-s + (−1.33 + 2.28i)14-s + (−0.583 + 3.82i)15-s − 1.00·16-s + (−1.32 + 4.94i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.414 + 0.910i)3-s + 0.500i·4-s + (0.962 − 0.272i)5-s + (0.662 − 0.247i)6-s + (−0.254 − 0.967i)7-s + (0.250 − 0.250i)8-s + (−0.656 − 0.754i)9-s + (−0.617 − 0.344i)10-s + (−1.24 − 0.719i)11-s + (−0.455 − 0.207i)12-s + (−0.361 + 0.0967i)13-s + (−0.356 + 0.610i)14-s + (−0.150 + 0.988i)15-s − 0.250·16-s + (−0.321 + 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.782 + 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.782 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.782 + 0.623i$
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: \((2,\ 630,\ (\ :1/2),\ -0.782 + 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.145403 - 0.415949i\)
\(L(\frac12)\) \(\approx\) \(0.145403 - 0.415949i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.717 - 1.57i)T \)
5 \( 1 + (-2.15 + 0.609i)T \)
7 \( 1 + (0.673 + 2.55i)T \)
good11 \( 1 + (4.13 + 2.38i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.30 - 0.348i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (1.32 - 4.94i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.62 + 0.939i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.99 + 1.33i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (4.90 + 8.48i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.30T + 31T^{2} \)
37 \( 1 + (-0.910 - 3.39i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (4.61 + 2.66i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.100 - 0.375i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (1.56 + 1.56i)T + 47iT^{2} \)
53 \( 1 + (0.0971 + 0.0260i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 - 1.49T + 59T^{2} \)
61 \( 1 - 13.4T + 61T^{2} \)
67 \( 1 + (-2.59 + 2.59i)T - 67iT^{2} \)
71 \( 1 + 14.1iT - 71T^{2} \)
73 \( 1 + (1.48 - 5.55i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + 3.85iT - 79T^{2} \)
83 \( 1 + (4.36 + 1.17i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (-7.07 + 12.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.9 - 2.94i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.24772733310436396992567013328, −9.759902679641218931749638605308, −8.710646059830083419332268517824, −7.910332626959059574878598905512, −6.48264105263249184387899303056, −5.67106552280233983901701271021, −4.54400032543567665267900524266, −3.55874202860239686470454540000, −2.19333624683997351076464193978, −0.27185709686696116417853330878, 1.91768970992265386695174259541, 2.65740528970197270966901702672, 5.23745460187408052182455289853, 5.47029204247423767502678915987, 6.61674601847682756610626175765, 7.26316580272590067278724416888, 8.201380826540791073465465161422, 9.200861636108819588429948053502, 9.957631858506332097072957046380, 10.80594523509154038979825589731

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