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Properties

Label 2-1512-9.4-c1-0-2
Degree $2$
Conductor $1512$
Sign $-0.855 - 0.517i$
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  + (0.164 − 0.284i)5-s + (0.5 + 0.866i)7-s + (−0.664 − 1.15i)11-s + (−1.53 + 2.66i)13-s − 7.35·17-s − 2.93·19-s + (−3.34 + 5.79i)23-s + (2.44 + 4.23i)25-s + (−3.88 − 6.72i)29-s + (1.63 − 2.83i)31-s + 0.329·35-s + 0.329·37-s + (0.135 − 0.234i)41-s + (5.48 + 9.49i)43-s + (0.571 + 0.989i)47-s + ⋯
L(s)  = 1  + (0.0735 − 0.127i)5-s + (0.188 + 0.327i)7-s + (−0.200 − 0.347i)11-s + (−0.426 + 0.739i)13-s − 1.78·17-s − 0.673·19-s + (−0.697 + 1.20i)23-s + (0.489 + 0.847i)25-s + (−0.720 − 1.24i)29-s + (0.293 − 0.508i)31-s + 0.0556·35-s + 0.0540·37-s + (0.0211 − 0.0366i)41-s + (0.836 + 1.44i)43-s + (0.0832 + 0.144i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 - 0.517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.855 - 0.517i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.855 - 0.517i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4989060104\)
\(L(\frac12)\) \(\approx\) \(0.4989060104\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.164 + 0.284i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.664 + 1.15i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.53 - 2.66i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 7.35T + 17T^{2} \)
19 \( 1 + 2.93T + 19T^{2} \)
23 \( 1 + (3.34 - 5.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.88 + 6.72i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.63 + 2.83i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.329T + 37T^{2} \)
41 \( 1 + (-0.135 + 0.234i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.48 - 9.49i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.571 - 0.989i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6.42T + 53T^{2} \)
59 \( 1 + (-0.372 + 0.644i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.42 + 7.65i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.28 - 7.42i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.60T + 71T^{2} \)
73 \( 1 + 13.4T + 73T^{2} \)
79 \( 1 + (-0.628 - 1.08i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.0316 + 0.0548i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 11.3T + 89T^{2} \)
97 \( 1 + (-5.51 - 9.55i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.501874948852345248825580143321, −9.184939445581428059531783056524, −8.214606209953547638762797410367, −7.46301908383803487555977707120, −6.47416898201925401016644853203, −5.79763504865198789679576925281, −4.70551567011665038656246688836, −4.02704699676859783141300458745, −2.65132931422994524601180877690, −1.74665088151753421873110051210, 0.18007897211395220512436647417, 1.95925756458330994563788132920, 2.86706237907641222765789787290, 4.23170324793711605611624182249, 4.78099555882421032471928864881, 5.93784839517486520509577548676, 6.78323478962213696558648313671, 7.43975728928448534272867213462, 8.508602048290398098113639469959, 8.935679700655729473738853784345

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