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Properties

Label 2-630-315.209-c1-0-1
Degree $2$
Conductor $630$
Sign $-0.324 - 0.946i$
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 + (−1.73 + 0.0696i)3-s + (−0.499 − 0.866i)4-s + (−2.09 + 0.790i)5-s + (0.805 − 1.53i)6-s + (−0.165 − 2.64i)7-s + 0.999·8-s + (2.99 − 0.240i)9-s + (0.360 − 2.20i)10-s + (1.16 + 0.673i)11-s + (0.925 + 1.46i)12-s + (−1.23 − 2.13i)13-s + (2.36 + 1.17i)14-s + (3.56 − 1.51i)15-s + (−0.5 + 0.866i)16-s + 0.246i·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.999 + 0.0401i)3-s + (−0.249 − 0.433i)4-s + (−0.935 + 0.353i)5-s + (0.328 − 0.626i)6-s + (−0.0627 − 0.998i)7-s + 0.353·8-s + (0.996 − 0.0803i)9-s + (0.114 − 0.697i)10-s + (0.351 + 0.202i)11-s + (0.267 + 0.422i)12-s + (−0.341 − 0.591i)13-s + (0.633 + 0.314i)14-s + (0.920 − 0.390i)15-s + (−0.125 + 0.216i)16-s + 0.0597i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.324 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.324 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.324 - 0.946i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.324 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.293899 + 0.411379i\)
\(L(\frac12)\) \(\approx\) \(0.293899 + 0.411379i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 - 0.0696i)T \)
5 \( 1 + (2.09 - 0.790i)T \)
7 \( 1 + (0.165 + 2.64i)T \)
good11 \( 1 + (-1.16 - 0.673i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.23 + 2.13i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.246iT - 17T^{2} \)
19 \( 1 - 1.47iT - 19T^{2} \)
23 \( 1 + (-4.15 - 7.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.56 + 2.63i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (8.15 - 4.71i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.87iT - 37T^{2} \)
41 \( 1 + (-4.73 - 8.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-7.27 - 4.20i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-8.27 - 4.77i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.10T + 53T^{2} \)
59 \( 1 + (2.66 + 4.61i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.46 - 3.73i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.74 - 1.00i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 15.3iT - 71T^{2} \)
73 \( 1 - 5.92T + 73T^{2} \)
79 \( 1 + (-0.270 + 0.467i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.04 - 0.603i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 + (-5.12 + 8.88i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.99258587908390676976374694919, −10.04594409936739686107043332490, −9.246616093378560364231978693539, −7.67615282594148891581931035834, −7.47169072092009685477149748688, −6.55692012921623022073728038061, −5.50096189647998634353310605115, −4.46781820068250217064442876584, −3.55397834931004761144107016783, −1.09202673416507354937250854416, 0.44167423606676384393860882569, 2.15299826155901849405394684176, 3.73885692750821665861867468611, 4.70039837459792572312311566809, 5.64333767626505083983167984951, 6.86449512484875688020942484296, 7.67094035281807627693730613303, 9.009594273417520299640812787334, 9.209690847595333767699435687294, 10.77330598032454660020774875563

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