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Properties

Label 2-630-5.2-c2-0-19
Degree $2$
Conductor $630$
Sign $0.423 + 0.905i$
Analytic cond. $17.1662$
Root an. cond. $4.14321$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + 2i·4-s + (0.578 − 4.96i)5-s + (1.87 + 1.87i)7-s + (2 − 2i)8-s + (−5.54 + 4.38i)10-s + 12.4·11-s + (3.13 − 3.13i)13-s − 3.74i·14-s − 4·16-s + (5.80 + 5.80i)17-s + 26.5i·19-s + (9.93 + 1.15i)20-s + (−12.4 − 12.4i)22-s + (10.4 − 10.4i)23-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + 0.5i·4-s + (0.115 − 0.993i)5-s + (0.267 + 0.267i)7-s + (0.250 − 0.250i)8-s + (−0.554 + 0.438i)10-s + 1.12·11-s + (0.241 − 0.241i)13-s − 0.267i·14-s − 0.250·16-s + (0.341 + 0.341i)17-s + 1.39i·19-s + (0.496 + 0.0578i)20-s + (−0.563 − 0.563i)22-s + (0.453 − 0.453i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.423 + 0.905i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.423 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.423 + 0.905i$
Analytic conductor: \(17.1662\)
Root analytic conductor: \(4.14321\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1),\ 0.423 + 0.905i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.592074567\)
\(L(\frac12)\) \(\approx\) \(1.592074567\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 \)
5 \( 1 + (-0.578 + 4.96i)T \)
7 \( 1 + (-1.87 - 1.87i)T \)
good11 \( 1 - 12.4T + 121T^{2} \)
13 \( 1 + (-3.13 + 3.13i)T - 169iT^{2} \)
17 \( 1 + (-5.80 - 5.80i)T + 289iT^{2} \)
19 \( 1 - 26.5iT - 361T^{2} \)
23 \( 1 + (-10.4 + 10.4i)T - 529iT^{2} \)
29 \( 1 + 14.5iT - 841T^{2} \)
31 \( 1 - 42.6T + 961T^{2} \)
37 \( 1 + (-11.9 - 11.9i)T + 1.36e3iT^{2} \)
41 \( 1 + 37.5T + 1.68e3T^{2} \)
43 \( 1 + (-24.0 + 24.0i)T - 1.84e3iT^{2} \)
47 \( 1 + (8.83 + 8.83i)T + 2.20e3iT^{2} \)
53 \( 1 + (-1.97 + 1.97i)T - 2.80e3iT^{2} \)
59 \( 1 + 88.2iT - 3.48e3T^{2} \)
61 \( 1 - 102.T + 3.72e3T^{2} \)
67 \( 1 + (22.8 + 22.8i)T + 4.48e3iT^{2} \)
71 \( 1 - 10.7T + 5.04e3T^{2} \)
73 \( 1 + (-80.4 + 80.4i)T - 5.32e3iT^{2} \)
79 \( 1 - 138. iT - 6.24e3T^{2} \)
83 \( 1 + (-96.0 + 96.0i)T - 6.88e3iT^{2} \)
89 \( 1 - 3.29iT - 7.92e3T^{2} \)
97 \( 1 + (88.5 + 88.5i)T + 9.40e3iT^{2} \)
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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

−10.03572481310891769205862426043, −9.459099024796960224606511530992, −8.402677578496521370089066446837, −8.126870436008729881936026813864, −6.65179353852065300436220053312, −5.65906949442978608503195211075, −4.48944622699416837212347540867, −3.53868992675985614948989056078, −1.92023159179012391733251876030, −0.909059143132960052178828293597, 1.11397902655087902002583975050, 2.66500852606784328329415392371, 3.96772754945140592265673654225, 5.18738128296397437303388701591, 6.45100196568883171586263078048, 6.88967460598027322869867034088, 7.78503079668111022214049534428, 8.892108626245810187149982288664, 9.577178634420890535160246348317, 10.48206674559466713384554791244

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