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Properties

Label 2-630-5.2-c2-0-0
Degree $2$
Conductor $630$
Sign $-0.452 + 0.891i$
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 + (−4.26 + 2.60i)5-s + (1.87 + 1.87i)7-s + (−2 + 2i)8-s + (−6.87 − 1.66i)10-s − 11.5·11-s + (−2.01 + 2.01i)13-s + 3.74i·14-s − 4·16-s + (−7.75 − 7.75i)17-s − 21.8i·19-s + (−5.20 − 8.53i)20-s + (−11.5 − 11.5i)22-s + (7.41 − 7.41i)23-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + 0.5i·4-s + (−0.853 + 0.520i)5-s + (0.267 + 0.267i)7-s + (−0.250 + 0.250i)8-s + (−0.687 − 0.166i)10-s − 1.05·11-s + (−0.155 + 0.155i)13-s + 0.267i·14-s − 0.250·16-s + (−0.456 − 0.456i)17-s − 1.14i·19-s + (−0.260 − 0.426i)20-s + (−0.526 − 0.526i)22-s + (0.322 − 0.322i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 + 0.891i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.452 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.452 + 0.891i$
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: \((2,\ 630,\ (\ :1),\ -0.452 + 0.891i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.004399047048\)
\(L(\frac12)\) \(\approx\) \(0.004399047048\)
\(L(2)\) 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 \( 1 + (4.26 - 2.60i)T \)
7 \( 1 + (-1.87 - 1.87i)T \)
good11 \( 1 + 11.5T + 121T^{2} \)
13 \( 1 + (2.01 - 2.01i)T - 169iT^{2} \)
17 \( 1 + (7.75 + 7.75i)T + 289iT^{2} \)
19 \( 1 + 21.8iT - 361T^{2} \)
23 \( 1 + (-7.41 + 7.41i)T - 529iT^{2} \)
29 \( 1 - 31.0iT - 841T^{2} \)
31 \( 1 + 11.5T + 961T^{2} \)
37 \( 1 + (32.0 + 32.0i)T + 1.36e3iT^{2} \)
41 \( 1 + 39.6T + 1.68e3T^{2} \)
43 \( 1 + (19.3 - 19.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (21.8 + 21.8i)T + 2.20e3iT^{2} \)
53 \( 1 + (-42.4 + 42.4i)T - 2.80e3iT^{2} \)
59 \( 1 + 89.0iT - 3.48e3T^{2} \)
61 \( 1 - 7.07T + 3.72e3T^{2} \)
67 \( 1 + (-15.6 - 15.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 133.T + 5.04e3T^{2} \)
73 \( 1 + (92.3 - 92.3i)T - 5.32e3iT^{2} \)
79 \( 1 - 126. iT - 6.24e3T^{2} \)
83 \( 1 + (-30.0 + 30.0i)T - 6.88e3iT^{2} \)
89 \( 1 + 4.93iT - 7.92e3T^{2} \)
97 \( 1 + (30.2 + 30.2i)T + 9.40e3iT^{2} \)
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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

−11.13988276000599371494746796760, −10.29383532327465734194305274602, −8.939358040613727091866184130582, −8.249203298764064138971453203034, −7.22599907959291371018275975147, −6.77676024223431926148681846708, −5.33309844303769281477662255420, −4.68718248255576590855980135621, −3.42613907859056958457181376855, −2.45130669807819338512245276901, 0.00131546471302446407628864655, 1.61009590517147294921585717198, 3.10359574562654685323899546741, 4.12560920607142318356616429780, 4.96535421544846622872411075324, 5.89986362764135003566155018424, 7.27547311682702498559642081687, 8.045678654666560011777692701966, 8.865827421484786272189447761231, 10.14918856617480340610563201101

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