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Properties

Label 2-630-21.20-c1-0-12
Degree $2$
Conductor $630$
Sign $0.996 + 0.0775i$
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 − 4-s + 5-s + (2.27 − 1.35i)7-s − i·8-s + i·10-s − 2.71i·11-s − 6.54i·13-s + (1.35 + 2.27i)14-s + 16-s − 1.53·17-s − 2.30i·19-s − 20-s + 2.71·22-s + 3.83i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.447·5-s + (0.858 − 0.512i)7-s − 0.353i·8-s + 0.316i·10-s − 0.817i·11-s − 1.81i·13-s + (0.362 + 0.607i)14-s + 0.250·16-s − 0.371·17-s − 0.528i·19-s − 0.223·20-s + 0.577·22-s + 0.799i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.996 + 0.0775i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.996 + 0.0775i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58678 - 0.0616224i\)
\(L(\frac12)\) \(\approx\) \(1.58678 - 0.0616224i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
7 \( 1 + (-2.27 + 1.35i)T \)
good11 \( 1 + 2.71iT - 11T^{2} \)
13 \( 1 + 6.54iT - 13T^{2} \)
17 \( 1 + 1.53T + 17T^{2} \)
19 \( 1 + 2.30iT - 19T^{2} \)
23 \( 1 - 3.83iT - 23T^{2} \)
29 \( 1 - 3.83iT - 29T^{2} \)
31 \( 1 + 3.25iT - 31T^{2} \)
37 \( 1 - 3.01T + 37T^{2} \)
41 \( 1 - 6.54T + 41T^{2} \)
43 \( 1 + 0.468T + 43T^{2} \)
47 \( 1 - 9.11T + 47T^{2} \)
53 \( 1 - 9.25iT - 53T^{2} \)
59 \( 1 + 11.1T + 59T^{2} \)
61 \( 1 - 4.78iT - 61T^{2} \)
67 \( 1 - 13.5T + 67T^{2} \)
71 \( 1 + 2.30iT - 71T^{2} \)
73 \( 1 - 11.4iT - 73T^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 + 13.0T + 83T^{2} \)
89 \( 1 - 9.60T + 89T^{2} \)
97 \( 1 - 16.9iT - 97T^{2} \)
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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

−10.67553333912311628649710225453, −9.618453585670484234021740213277, −8.648520717108896113642983762995, −7.896284462240835988296221386834, −7.17448402019268136502817880113, −5.88065675639661628084396138839, −5.35554636463662306049542291161, −4.18904622420341572432611977933, −2.86320162444804757165269493763, −0.962197916435559099262729242341, 1.68579375936221524165269925593, 2.41478031577196321844226474231, 4.16626174504471507713025296981, 4.78738770347956771717379591495, 6.00652697282938243271070296285, 7.04901420307976278903284502425, 8.218271661451879230792493672227, 9.084810031332554557960478079271, 9.686091601498862138526559747168, 10.70771645972984775339108253218

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