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Properties

Label 2-1512-63.4-c1-0-15
Degree $2$
Conductor $1512$
Sign $0.367 + 0.929i$
Analytic cond. $12.0733$
Root an. cond. $3.47467$
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  + 0.0619·5-s + (−1.63 − 2.07i)7-s + 3.18·11-s + (−0.252 + 0.437i)13-s + (0.554 − 0.960i)17-s + (0.933 + 1.61i)19-s + 6.20·23-s − 4.99·25-s + (−2.39 − 4.15i)29-s + (1.26 + 2.19i)31-s + (−0.101 − 0.128i)35-s + (−4.26 − 7.38i)37-s + (4.94 − 8.56i)41-s + (−3.95 − 6.85i)43-s + (3.29 − 5.70i)47-s + ⋯
L(s)  = 1  + 0.0277·5-s + (−0.618 − 0.785i)7-s + 0.958·11-s + (−0.0700 + 0.121i)13-s + (0.134 − 0.233i)17-s + (0.214 + 0.370i)19-s + 1.29·23-s − 0.999·25-s + (−0.445 − 0.770i)29-s + (0.227 + 0.394i)31-s + (−0.0171 − 0.0217i)35-s + (−0.700 − 1.21i)37-s + (0.772 − 1.33i)41-s + (−0.603 − 1.04i)43-s + (0.480 − 0.831i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.367 + 0.929i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.367 + 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.484356612\)
\(L(\frac12)\) \(\approx\) \(1.484356612\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (1.63 + 2.07i)T \)
good5 \( 1 - 0.0619T + 5T^{2} \)
11 \( 1 - 3.18T + 11T^{2} \)
13 \( 1 + (0.252 - 0.437i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.554 + 0.960i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.933 - 1.61i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.20T + 23T^{2} \)
29 \( 1 + (2.39 + 4.15i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.26 - 2.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.26 + 7.38i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.94 + 8.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.95 + 6.85i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.29 + 5.70i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.58 + 2.74i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.50 - 7.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.94 + 12.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.66 + 2.88i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.25T + 71T^{2} \)
73 \( 1 + (-2.07 + 3.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.48 + 2.57i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.17 + 3.76i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.30 - 7.44i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.27 + 5.67i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.334467202364631631067776224972, −8.690151033242274075430732415419, −7.47169164311639562948486938416, −7.02655234968248553029704064885, −6.12450337496148383180816387629, −5.20947437858441343263627203887, −3.99549206856270953566596762722, −3.50142940015882614569056613940, −2.05761836825868834357270904323, −0.64804161985996305784695433937, 1.28938812547933101981539666606, 2.67245574877995004574672346596, 3.51199294344181309159332705882, 4.63090291805256756929625199169, 5.60333349639252891682877565822, 6.38052389083544820770427108043, 7.08106711790885165335162953772, 8.130224849917349951439388411005, 8.973523696613763248480595107963, 9.500804172279479695573223974010

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