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Properties

Label 2-630-9.4-c1-0-13
Degree $2$
Conductor $630$
Sign $-0.514 + 0.857i$
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  + (−0.5 − 0.866i)2-s + (−1.62 − 0.606i)3-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.285 + 1.70i)6-s + (−0.5 − 0.866i)7-s + 0.999·8-s + (2.26 + 1.96i)9-s + 0.999·10-s + (0.285 + 0.495i)11-s + (1.33 − 1.10i)12-s + (0.214 − 0.370i)13-s + (−0.499 + 0.866i)14-s + (1.33 − 1.10i)15-s + (−0.5 − 0.866i)16-s + 5.67·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.936 − 0.350i)3-s + (−0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s + (0.116 + 0.697i)6-s + (−0.188 − 0.327i)7-s + 0.353·8-s + (0.754 + 0.655i)9-s + 0.316·10-s + (0.0862 + 0.149i)11-s + (0.385 − 0.318i)12-s + (0.0593 − 0.102i)13-s + (−0.133 + 0.231i)14-s + (0.345 − 0.284i)15-s + (−0.125 − 0.216i)16-s + 1.37·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.514 + 0.857i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.514 + 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.319987 - 0.565478i\)
\(L(\frac12)\) \(\approx\) \(0.319987 - 0.565478i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (1.62 + 0.606i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good11 \( 1 + (-0.285 - 0.495i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.214 + 0.370i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.67T + 17T^{2} \)
19 \( 1 + 4.81T + 19T^{2} \)
23 \( 1 + (-3.88 + 6.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.45 + 7.72i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.24 - 7.35i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.95T + 37T^{2} \)
41 \( 1 + (-5.63 + 9.75i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.07 + 5.32i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.67 + 9.82i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 5.10T + 53T^{2} \)
59 \( 1 + (3.16 - 5.47i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.12 - 3.67i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.65 + 9.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 15.0T + 71T^{2} \)
73 \( 1 + 6.34T + 73T^{2} \)
79 \( 1 + (3.67 + 6.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.10 + 8.83i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 8.28T + 89T^{2} \)
97 \( 1 + (2.74 + 4.75i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.53248626844357689851222355754, −9.772552539078589721151113987519, −8.554765220071692658375231655074, −7.57476168699659462594988372322, −6.83131520020228065369578888933, −5.82113491099055763460169184468, −4.63715237655820098810056590078, −3.56509621424143182568976880650, −2.09092974584917809848809437065, −0.51704033102637759674729286783, 1.26204329911173902688392302641, 3.50560747064375702568770270132, 4.69062660340858138790038179825, 5.57590387606129418951799714642, 6.25134858778211103861710802403, 7.34206023909293337576667833147, 8.173056648419013795941762042619, 9.484891552585673337616387637332, 9.626842136691553483730101549107, 11.08972795968646726108700780065

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