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Properties

Label 2-630-63.38-c1-0-27
Degree $2$
Conductor $630$
Sign $-0.373 + 0.927i$
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  − i·2-s + (0.939 − 1.45i)3-s − 4-s + (−0.5 + 0.866i)5-s + (−1.45 − 0.939i)6-s + (2.57 + 0.617i)7-s + i·8-s + (−1.23 − 2.73i)9-s + (0.866 + 0.5i)10-s + (2.66 − 1.53i)11-s + (−0.939 + 1.45i)12-s + (0.449 − 0.259i)13-s + (0.617 − 2.57i)14-s + (0.790 + 1.54i)15-s + 16-s + (2.44 − 4.23i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.542 − 0.840i)3-s − 0.5·4-s + (−0.223 + 0.387i)5-s + (−0.594 − 0.383i)6-s + (0.972 + 0.233i)7-s + 0.353i·8-s + (−0.411 − 0.911i)9-s + (0.273 + 0.158i)10-s + (0.803 − 0.463i)11-s + (−0.271 + 0.420i)12-s + (0.124 − 0.0720i)13-s + (0.164 − 0.687i)14-s + (0.204 + 0.397i)15-s + 0.250·16-s + (0.592 − 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.373 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.373 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.373 + 0.927i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.373 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00386 - 1.48640i\)
\(L(\frac12)\) \(\approx\) \(1.00386 - 1.48640i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-0.939 + 1.45i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-2.57 - 0.617i)T \)
good11 \( 1 + (-2.66 + 1.53i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.449 + 0.259i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.44 + 4.23i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.713 + 0.412i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.74 + 4.47i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.55 - 3.20i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.905iT - 31T^{2} \)
37 \( 1 + (-2.53 - 4.39i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.18 + 3.78i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.84 - 3.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.78T + 47T^{2} \)
53 \( 1 + (4.69 + 2.71i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 - 9.52iT - 61T^{2} \)
67 \( 1 - 7.90T + 67T^{2} \)
71 \( 1 - 10.6iT - 71T^{2} \)
73 \( 1 + (-5.78 - 3.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 3.46T + 79T^{2} \)
83 \( 1 + (-3.32 + 5.75i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.771 + 1.33i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.93 + 1.11i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.41238835635327793650592439858, −9.402009028061277440477847372331, −8.477842076972441294427316462244, −7.941243462285531787808936561362, −6.86680029462126689606813859350, −5.83981453036712675259665886752, −4.49962432632472348317011247356, −3.33714364317872848544107086665, −2.32487008231116775647269717130, −1.06031726173633999783707258745, 1.73220595739289301723379788640, 3.69576247729004685923499466798, 4.33054240824941016156749345528, 5.25497827644715957628073680512, 6.29909173661194314883293071162, 7.83249142308808245922939389943, 8.008928995603407925121517961068, 9.060139377108947374717869826999, 9.807422183784231193061598087339, 10.65445232542141532269774118591

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