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Properties

Label 2-1512-7.4-c1-0-13
Degree $2$
Conductor $1512$
Sign $0.991 - 0.126i$
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  + (−1 − 1.73i)5-s + (2.5 + 0.866i)7-s + (−1.5 + 2.59i)11-s + 6·13-s + (−2 + 3.46i)17-s + (−2 − 3.46i)19-s + (2 + 3.46i)23-s + (0.500 − 0.866i)25-s + 5·29-s + (−3.5 + 6.06i)31-s + (−1.00 − 5.19i)35-s − 2·41-s + 8·43-s + (−1 − 1.73i)47-s + (5.5 + 4.33i)49-s + ⋯
L(s)  = 1  + (−0.447 − 0.774i)5-s + (0.944 + 0.327i)7-s + (−0.452 + 0.783i)11-s + 1.66·13-s + (−0.485 + 0.840i)17-s + (−0.458 − 0.794i)19-s + (0.417 + 0.722i)23-s + (0.100 − 0.173i)25-s + 0.928·29-s + (−0.628 + 1.08i)31-s + (−0.169 − 0.878i)35-s − 0.312·41-s + 1.21·43-s + (−0.145 − 0.252i)47-s + (0.785 + 0.618i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.991 - 0.126i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.790519020\)
\(L(\frac12)\) \(\approx\) \(1.790519020\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 0.866i)T \)
good5 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5 + 8.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3 + 5.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 15T + 83T^{2} \)
89 \( 1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5T + 97T^{2} \)
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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

−9.177803805478452960543132314568, −8.597061421258669667222882239348, −8.160354777585803268312950093429, −7.15234666870312418920262191525, −6.18958289382464225742915066893, −5.16364083859426110812322377308, −4.54989270091038728997487255339, −3.64573618353519178857988394556, −2.17396195477479501661724827651, −1.10755901281562975072652870162, 0.915592310022598177256712504551, 2.39594649831669469681774526345, 3.49481735598403349573924349875, 4.24908049407290959179895575174, 5.36511006869291827397666501949, 6.26793757073321076773438340087, 7.04484795662703222322487643593, 8.032677717087618361400923397047, 8.400638206995269276785213430887, 9.376694052532446034408918411266

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