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Properties

Label 2-630-105.23-c1-0-0
Degree $2$
Conductor $630$
Sign $-0.862 + 0.506i$
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  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−1.30 + 1.81i)5-s + (0.634 + 2.56i)7-s + (−0.707 + 0.707i)8-s + (0.786 − 2.09i)10-s + (−4.28 + 2.47i)11-s + (−4.34 − 4.34i)13-s + (−1.27 − 2.31i)14-s + (0.500 − 0.866i)16-s + (1.13 − 4.25i)17-s + (3.22 + 1.86i)19-s + (−0.218 + 2.22i)20-s + (3.49 − 3.49i)22-s + (−0.503 − 1.87i)23-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.433 − 0.249i)4-s + (−0.582 + 0.813i)5-s + (0.239 + 0.970i)7-s + (−0.249 + 0.249i)8-s + (0.248 − 0.661i)10-s + (−1.29 + 0.745i)11-s + (−1.20 − 1.20i)13-s + (−0.341 − 0.619i)14-s + (0.125 − 0.216i)16-s + (0.276 − 1.03i)17-s + (0.740 + 0.427i)19-s + (−0.0487 + 0.497i)20-s + (0.745 − 0.745i)22-s + (−0.104 − 0.391i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.862 + 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.862 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.862 + 0.506i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.862 + 0.506i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0309575 - 0.113739i\)
\(L(\frac12)\) \(\approx\) \(0.0309575 - 0.113739i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 \)
5 \( 1 + (1.30 - 1.81i)T \)
7 \( 1 + (-0.634 - 2.56i)T \)
good11 \( 1 + (4.28 - 2.47i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.34 + 4.34i)T + 13iT^{2} \)
17 \( 1 + (-1.13 + 4.25i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-3.22 - 1.86i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.503 + 1.87i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 3.56T + 29T^{2} \)
31 \( 1 + (0.272 + 0.471i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.435 - 1.62i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.268iT - 41T^{2} \)
43 \( 1 + (8.39 + 8.39i)T + 43iT^{2} \)
47 \( 1 + (0.896 - 0.240i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (7.70 + 2.06i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-4.90 - 8.48i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.71 - 9.89i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.0408 - 0.0109i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 16.3iT - 71T^{2} \)
73 \( 1 + (2.25 - 8.42i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-6.93 - 4.00i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.30 - 1.30i)T - 83iT^{2} \)
89 \( 1 + (0.229 - 0.397i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.6 - 10.6i)T - 97iT^{2} \)
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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

−10.90184981134583127343267936033, −10.13509346528936571363863316019, −9.552090920790255386592254696175, −8.224955326642094389699304856062, −7.66831684378898536154877656991, −7.03402496746128208439117984502, −5.61890843103440296705477925431, −4.96240433456270535084633168478, −3.05660455032349568407209539890, −2.36068396647357723448355417382, 0.07674403230848402623738219888, 1.63380882373565521425032619940, 3.27116493622432450674612135341, 4.43497906435858396024375782088, 5.35989257125450109344651642207, 6.82733509867980518747873612685, 7.76833229258410635448359430554, 8.141669622295521606936241130800, 9.309532604659502822588770856346, 9.980613995440552274627712769172

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