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Properties

Label 2-1512-63.47-c1-0-4
Degree $2$
Conductor $1512$
Sign $-0.351 - 0.936i$
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  − 2.22·5-s + (2.51 − 0.814i)7-s − 2.12i·11-s + (−5.10 + 2.94i)13-s + (2.34 + 4.05i)17-s + (−4.54 − 2.62i)19-s + 4.36i·23-s − 0.0346·25-s + (2.25 + 1.30i)29-s + (6.59 + 3.80i)31-s + (−5.60 + 1.81i)35-s + (−1.80 + 3.12i)37-s + (−0.0395 − 0.0684i)41-s + (−1.24 + 2.16i)43-s + (−1.89 − 3.28i)47-s + ⋯
L(s)  = 1  − 0.996·5-s + (0.951 − 0.307i)7-s − 0.639i·11-s + (−1.41 + 0.817i)13-s + (0.567 + 0.983i)17-s + (−1.04 − 0.602i)19-s + 0.909i·23-s − 0.00693·25-s + (0.418 + 0.241i)29-s + (1.18 + 0.683i)31-s + (−0.948 + 0.306i)35-s + (−0.296 + 0.513i)37-s + (−0.00616 − 0.0106i)41-s + (−0.190 + 0.329i)43-s + (−0.276 − 0.479i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.351 - 0.936i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.351 - 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7744620694\)
\(L(\frac12)\) \(\approx\) \(0.7744620694\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.51 + 0.814i)T \)
good5 \( 1 + 2.22T + 5T^{2} \)
11 \( 1 + 2.12iT - 11T^{2} \)
13 \( 1 + (5.10 - 2.94i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.34 - 4.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.54 + 2.62i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.36iT - 23T^{2} \)
29 \( 1 + (-2.25 - 1.30i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.59 - 3.80i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.80 - 3.12i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.0395 + 0.0684i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.24 - 2.16i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.89 + 3.28i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.08 + 2.35i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.59 - 11.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.06 - 4.07i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.37 + 4.11i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.0iT - 71T^{2} \)
73 \( 1 + (12.6 - 7.30i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.27 - 12.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.41 - 11.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.73 - 4.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (12.9 + 7.46i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.787029969696830021807234400030, −8.618082759379813629327386373580, −8.200986229079734255352180779395, −7.37589481252450242638999948163, −6.68691842978301946317467035069, −5.46367148415060796898998454404, −4.54442726934764681094912341459, −3.96609036613989606292821024173, −2.72542941788027436470631349937, −1.38786653003419928585918592472, 0.31265240685125043036636361406, 2.06611017411125333345644185249, 3.04346858998630148644151545211, 4.45262658093536615939178443643, 4.75259621878330143754122221535, 5.86991721686295398766867279968, 7.03818690111580572358643450166, 7.86632472812125927957479726158, 8.068311845703466807450844521354, 9.201892659615237428558778033329

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