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Properties

Label 2-630-63.59-c1-0-13
Degree $2$
Conductor $630$
Sign $-0.258 - 0.966i$
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 + (0.0292 + 1.73i)3-s + (0.499 + 0.866i)4-s + 5-s + (−0.840 + 1.51i)6-s + (2.63 + 0.194i)7-s + 0.999i·8-s + (−2.99 + 0.101i)9-s + (0.866 + 0.5i)10-s + 2.09i·11-s + (−1.48 + 0.891i)12-s + (−0.413 − 0.238i)13-s + (2.18 + 1.48i)14-s + (0.0292 + 1.73i)15-s + (−0.5 + 0.866i)16-s + (−1.44 + 2.49i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.0168 + 0.999i)3-s + (0.249 + 0.433i)4-s + 0.447·5-s + (−0.343 + 0.618i)6-s + (0.997 + 0.0735i)7-s + 0.353i·8-s + (−0.999 + 0.0337i)9-s + (0.273 + 0.158i)10-s + 0.630i·11-s + (−0.428 + 0.257i)12-s + (−0.114 − 0.0662i)13-s + (0.584 + 0.397i)14-s + (0.00755 + 0.447i)15-s + (−0.125 + 0.216i)16-s + (−0.349 + 0.605i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.258 - 0.966i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.258 - 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44254 + 1.87928i\)
\(L(\frac12)\) \(\approx\) \(1.44254 + 1.87928i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0292 - 1.73i)T \)
5 \( 1 - T \)
7 \( 1 + (-2.63 - 0.194i)T \)
good11 \( 1 - 2.09iT - 11T^{2} \)
13 \( 1 + (0.413 + 0.238i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.44 - 2.49i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.03 + 2.32i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.02iT - 23T^{2} \)
29 \( 1 + (-2.74 + 1.58i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.59 - 2.65i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.93 + 3.34i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.03 - 6.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.96 + 5.14i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.0525 + 0.0909i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.27 + 2.46i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.40 - 2.44i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.71 - 2.14i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.73 + 4.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.8iT - 71T^{2} \)
73 \( 1 + (-5.64 - 3.26i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.81 + 13.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.58 - 9.68i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.59 + 11.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.98 + 4.60i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.85547320523706285255527346033, −10.06384554050094274794753615099, −9.074195781472060017067994189787, −8.296707505320967532835130065171, −7.26084386957075662369570688970, −6.10431982784578331494365919049, −5.09114570312642102691588503160, −4.62920623939126711867058344931, −3.41511366746878111593334161046, −2.09055837399166878201390789908, 1.19570272765458749265553912613, 2.27480498177709257868049898829, 3.47249443335335457880344311594, 5.01142229494001997659341395062, 5.65302765647583570950853242548, 6.70944902769173677529855084396, 7.59706416638103797723520768035, 8.483896203939299605875357489395, 9.480056527392162554560288967988, 10.65585089843323142032108193126

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