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Properties

Label 2-630-5.2-c2-0-16
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 − 19.5·11-s + (8.03 − 8.03i)13-s + 3.74i·14-s − 4·16-s + (2.19 + 2.19i)17-s − 8.25i·19-s + (−9.93 − 1.15i)20-s + (19.5 + 19.5i)22-s + (17.9 − 17.9i)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 − 1.77·11-s + (0.617 − 0.617i)13-s + 0.267i·14-s − 0.250·16-s + (0.129 + 0.129i)17-s − 0.434i·19-s + (−0.496 − 0.0578i)20-s + (0.889 + 0.889i)22-s + (0.778 − 0.778i)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

\(L(\frac{3}{2})\) \(\approx\) \(0.9860667135\)
\(L(\frac12)\) \(\approx\) \(0.9860667135\)
\(L(2)\) 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 \( 1 + 19.5T + 121T^{2} \)
13 \( 1 + (-8.03 + 8.03i)T - 169iT^{2} \)
17 \( 1 + (-2.19 - 2.19i)T + 289iT^{2} \)
19 \( 1 + 8.25iT - 361T^{2} \)
23 \( 1 + (-17.9 + 17.9i)T - 529iT^{2} \)
29 \( 1 + 19.7iT - 841T^{2} \)
31 \( 1 - 30.0T + 961T^{2} \)
37 \( 1 + (-37.2 - 37.2i)T + 1.36e3iT^{2} \)
41 \( 1 - 80.8T + 1.68e3T^{2} \)
43 \( 1 + (13.6 - 13.6i)T - 1.84e3iT^{2} \)
47 \( 1 + (-8.17 - 8.17i)T + 2.20e3iT^{2} \)
53 \( 1 + (-38.8 + 38.8i)T - 2.80e3iT^{2} \)
59 \( 1 + 74.3iT - 3.48e3T^{2} \)
61 \( 1 - 97.8T + 3.72e3T^{2} \)
67 \( 1 + (67.1 + 67.1i)T + 4.48e3iT^{2} \)
71 \( 1 - 13.3T + 5.04e3T^{2} \)
73 \( 1 + (48.2 - 48.2i)T - 5.32e3iT^{2} \)
79 \( 1 + 40.2iT - 6.24e3T^{2} \)
83 \( 1 + (34.4 - 34.4i)T - 6.88e3iT^{2} \)
89 \( 1 + 157. iT - 7.92e3T^{2} \)
97 \( 1 + (73.2 + 73.2i)T + 9.40e3iT^{2} \)
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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

−10.35037500871091366258038651828, −9.681774826836214633063415268896, −8.316834985035039434344348525709, −7.79844545172127265611743185741, −6.82021295300622071224835893918, −5.80152448714420234529437944787, −4.45878532976449513538895980445, −3.09626595148461535237268852004, −2.53356572108202867705926510965, −0.53954157189371384595256593147, 0.998468291549321042079229894182, 2.56013690395923923067013681644, 4.17652429706271797348102010852, 5.30906424423823056792020768711, 5.86644894930114924974649362642, 7.24011201630979477873638556141, 7.974392444556668288069835928106, 8.781858226672617774757565109532, 9.466142169338353414351393661867, 10.40456730199717584339302217356

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