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Properties

Label 2-630-315.194-c1-0-19
Degree $2$
Conductor $630$
Sign $-0.0156 - 0.999i$
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 + (0.630 + 1.61i)3-s + 4-s + (0.0611 + 2.23i)5-s + (0.630 + 1.61i)6-s + (1.83 + 1.90i)7-s + 8-s + (−2.20 + 2.03i)9-s + (0.0611 + 2.23i)10-s + (−1.13 − 0.652i)11-s + (0.630 + 1.61i)12-s + (3.03 − 5.25i)13-s + (1.83 + 1.90i)14-s + (−3.56 + 1.50i)15-s + 16-s + (0.502 − 0.290i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.363 + 0.931i)3-s + 0.5·4-s + (0.0273 + 0.999i)5-s + (0.257 + 0.658i)6-s + (0.693 + 0.720i)7-s + 0.353·8-s + (−0.735 + 0.677i)9-s + (0.0193 + 0.706i)10-s + (−0.340 − 0.196i)11-s + (0.181 + 0.465i)12-s + (0.841 − 1.45i)13-s + (0.490 + 0.509i)14-s + (−0.921 + 0.389i)15-s + 0.250·16-s + (0.121 − 0.0704i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0156 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0156 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.0156 - 0.999i$
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: \((2,\ 630,\ (\ :1/2),\ -0.0156 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84168 + 1.87064i\)
\(L(\frac12)\) \(\approx\) \(1.84168 + 1.87064i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.630 - 1.61i)T \)
5 \( 1 + (-0.0611 - 2.23i)T \)
7 \( 1 + (-1.83 - 1.90i)T \)
good11 \( 1 + (1.13 + 0.652i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.03 + 5.25i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.502 + 0.290i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.86 + 3.96i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.28 - 5.69i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.125 - 0.0724i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.38iT - 31T^{2} \)
37 \( 1 + (-2.02 - 1.17i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.16 + 5.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-10.0 + 5.81i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.41iT - 47T^{2} \)
53 \( 1 + (-4.19 - 7.25i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 3.08T + 59T^{2} \)
61 \( 1 - 2.12iT - 61T^{2} \)
67 \( 1 + 13.1iT - 67T^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + (-3.84 - 6.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 7.09T + 79T^{2} \)
83 \( 1 + (4.32 - 2.49i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.35 + 2.34i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.76 + 9.97i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.94212620895617449222823517832, −10.28418459352829052224231395131, −9.020626391089745532410523190749, −8.234967087206644337152820610032, −7.29900810396724052546912715267, −5.89949607411211948319245348141, −5.40457028445230362526774983477, −4.18298324632507652042414200340, −3.14541852521747078661060455037, −2.38258445618636341655045714739, 1.24454243176292667678245051206, 2.23583808209447259846863425234, 3.99902443865807611599118274087, 4.57848010993951260552489759779, 5.96083517741186879217018415524, 6.67958285755131458518629713337, 7.79133200164352818681342335359, 8.410213042413740945538972154763, 9.270584697324653107404451298776, 10.67430651858435328009734746293

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