Properties

Label 2-630-63.47-c1-0-13
Degree $2$
Conductor $630$
Sign $0.812 - 0.583i$
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  + (−0.866 + 0.5i)2-s + (1.70 + 0.311i)3-s + (0.499 − 0.866i)4-s − 5-s + (−1.63 + 0.582i)6-s + (2.53 + 0.744i)7-s + 0.999i·8-s + (2.80 + 1.06i)9-s + (0.866 − 0.5i)10-s − 0.441i·11-s + (1.12 − 1.32i)12-s + (3.17 − 1.83i)13-s + (−2.57 + 0.624i)14-s + (−1.70 − 0.311i)15-s + (−0.5 − 0.866i)16-s + (0.136 + 0.235i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.983 + 0.179i)3-s + (0.249 − 0.433i)4-s − 0.447·5-s + (−0.665 + 0.237i)6-s + (0.959 + 0.281i)7-s + 0.353i·8-s + (0.935 + 0.353i)9-s + (0.273 − 0.158i)10-s − 0.133i·11-s + (0.323 − 0.381i)12-s + (0.880 − 0.508i)13-s + (−0.687 + 0.166i)14-s + (−0.439 − 0.0803i)15-s + (−0.125 − 0.216i)16-s + (0.0330 + 0.0571i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.812 - 0.583i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.812 - 0.583i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58440 + 0.510062i\)
\(L(\frac12)\) \(\approx\) \(1.58440 + 0.510062i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-1.70 - 0.311i)T \)
5 \( 1 + T \)
7 \( 1 + (-2.53 - 0.744i)T \)
good11 \( 1 + 0.441iT - 11T^{2} \)
13 \( 1 + (-3.17 + 1.83i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.136 - 0.235i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.25 + 1.87i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.59iT - 23T^{2} \)
29 \( 1 + (2.38 + 1.37i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-7.57 - 4.37i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.0597 + 0.103i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.93 - 6.81i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.849 + 1.47i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.94 + 6.83i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.0822 + 0.0474i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.60 + 2.78i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (11.2 - 6.50i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.268 + 0.465i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.75iT - 71T^{2} \)
73 \( 1 + (-9.64 + 5.56i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.51 + 2.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.29 - 7.43i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.35 - 11.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (12.9 + 7.46i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.62614079071457939207794045151, −9.614299381130055509860052234285, −8.638728177194171948857926377984, −8.254551644556189155459903611727, −7.51669004003649085524608617917, −6.40624223458657046831824041542, −5.12416275333240595648390320053, −4.07838692068821322607476234384, −2.79272266734773033647505354908, −1.42460873702357336916485098648, 1.29283470923669305625254494568, 2.43458719457828878268522770159, 3.79581383774593265705115541992, 4.51098914844113087543254668080, 6.31174394308091598564569886090, 7.34575133221101300027028669768, 8.111053677252589737879858582358, 8.612461292531976255000707023178, 9.506482178145850694745075525631, 10.52228569573772046643429103911

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