Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Sign $0.812 + 0.583i$
Motivic weight 1
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

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.812 + 0.583i$
motivic weight  =  \(1\)
character  :  $\chi_{630} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 630,\ (\ :1/2),\ 0.812 + 0.583i)\)
\(L(1)\)  \(\approx\)  \(1.58440 - 0.510062i\)
\(L(\frac12)\)  \(\approx\)  \(1.58440 - 0.510062i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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