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Properties

Label 2-630-315.23-c1-0-19
Degree $2$
Conductor $630$
Sign $0.695 - 0.718i$
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 + (−0.913 + 1.47i)3-s + 1.00i·4-s + (0.187 + 2.22i)5-s + (1.68 − 0.394i)6-s + (1.77 − 1.96i)7-s + (0.707 − 0.707i)8-s + (−1.32 − 2.68i)9-s + (1.44 − 1.70i)10-s + (3.93 + 2.27i)11-s + (−1.47 − 0.913i)12-s + (4.22 − 1.13i)13-s + (−2.64 + 0.134i)14-s + (−3.44 − 1.76i)15-s − 1.00·16-s + (0.304 − 1.13i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.527 + 0.849i)3-s + 0.500i·4-s + (0.0838 + 0.996i)5-s + (0.688 − 0.160i)6-s + (0.670 − 0.742i)7-s + (0.250 − 0.250i)8-s + (−0.443 − 0.896i)9-s + (0.456 − 0.540i)10-s + (1.18 + 0.685i)11-s + (−0.424 − 0.263i)12-s + (1.17 − 0.313i)13-s + (−0.706 + 0.0358i)14-s + (−0.890 − 0.454i)15-s − 0.250·16-s + (0.0737 − 0.275i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.695 - 0.718i$
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: \((2,\ 630,\ (\ :1/2),\ 0.695 - 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03631 + 0.439244i\)
\(L(\frac12)\) \(\approx\) \(1.03631 + 0.439244i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.913 - 1.47i)T \)
5 \( 1 + (-0.187 - 2.22i)T \)
7 \( 1 + (-1.77 + 1.96i)T \)
good11 \( 1 + (-3.93 - 2.27i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.22 + 1.13i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-0.304 + 1.13i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (5.35 + 3.09i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.27 - 1.94i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-3.13 - 5.42i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.358T + 31T^{2} \)
37 \( 1 + (-2.46 - 9.20i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (8.04 + 4.64i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.19 - 8.19i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-2.15 - 2.15i)T + 47iT^{2} \)
53 \( 1 + (-0.0705 - 0.0189i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 - 8.50T + 59T^{2} \)
61 \( 1 + 5.89T + 61T^{2} \)
67 \( 1 + (-6.76 + 6.76i)T - 67iT^{2} \)
71 \( 1 - 13.2iT - 71T^{2} \)
73 \( 1 + (-2.45 + 9.16i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + 5.26iT - 79T^{2} \)
83 \( 1 + (-3.57 - 0.957i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (2.74 - 4.75i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.37 + 0.903i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.81420850939378876077584199880, −10.06372647889160585421164634841, −9.175029553176885134333117428348, −8.335487613209650437807949621647, −6.93877467086652014973517624120, −6.51944236766568937700505144941, −4.94321166703025153411787200841, −4.00281696820871334090777087008, −3.10921816761902925897647650586, −1.30461068576848548532609867509, 0.976298994075756634774445149106, 1.94695519847727068269459271136, 4.14017102741971416136358986760, 5.35026130916751825831111003907, 6.06240598556128257857921581170, 6.75779472520007606169324806461, 8.195550194800521552188508346605, 8.522530198300174565545954783494, 9.176250463813735921356366323327, 10.64450494376994542201410984718

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