Properties

Degree $2$
Conductor $1512$
Sign $0.100 + 0.994i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 − 2.36i)5-s + (−0.931 + 2.47i)7-s + (1.17 − 2.04i)11-s + 6.68·13-s + (−2.09 + 3.63i)17-s + (−1.80 − 3.11i)19-s + (−3.40 − 5.90i)23-s + (−1.23 + 2.13i)25-s + 1.00·29-s + (−1.05 + 1.82i)31-s + (7.13 − 1.17i)35-s + (2.47 + 4.29i)37-s + 8.91·41-s − 2.25·43-s + (−5.85 − 10.1i)47-s + ⋯
L(s)  = 1  + (−0.610 − 1.05i)5-s + (−0.352 + 0.935i)7-s + (0.355 − 0.615i)11-s + 1.85·13-s + (−0.508 + 0.881i)17-s + (−0.413 − 0.715i)19-s + (−0.710 − 1.23i)23-s + (−0.246 + 0.426i)25-s + 0.186·29-s + (−0.189 + 0.327i)31-s + (1.20 − 0.199i)35-s + (0.407 + 0.706i)37-s + 1.39·41-s − 0.343·43-s + (−0.853 − 1.47i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.100 + 0.994i$
Motivic weight: \(1\)
Character: $\chi_{1512} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.100 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.284338879\)
\(L(\frac12)\) \(\approx\) \(1.284338879\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.931 - 2.47i)T \)
good5 \( 1 + (1.36 + 2.36i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.17 + 2.04i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.68T + 13T^{2} \)
17 \( 1 + (2.09 - 3.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.80 + 3.11i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.40 + 5.90i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.00T + 29T^{2} \)
31 \( 1 + (1.05 - 1.82i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.47 - 4.29i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.91T + 41T^{2} \)
43 \( 1 + 2.25T + 43T^{2} \)
47 \( 1 + (5.85 + 10.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.13 + 7.17i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.91 + 10.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.00 + 10.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.03 + 5.25i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.89T + 71T^{2} \)
73 \( 1 + (-2.28 + 3.96i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.252 - 0.437i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.1T + 83T^{2} \)
89 \( 1 + (4.22 + 7.32i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.25T + 97T^{2} \)
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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

−8.872408664050053671688755500718, −8.562195210860373072009955495733, −8.173280804713164946944073770485, −6.49318059393040889104319871324, −6.20086656113431057517612721243, −5.09142407537371481318564055760, −4.15693982151545151473460602391, −3.37914285349999702716660866080, −1.95071449623415159785052764685, −0.56900562117707439395768722492, 1.27169706354273796400999819832, 2.80341582084159481311282125308, 3.87321869016964869947796854362, 4.15020927828518335270759860590, 5.81461962726072968447592803368, 6.49738423278336743718788122302, 7.31796962487070822110813129716, 7.78728553212197486647417747605, 8.908360209222802149740581748126, 9.744246821860300594628772352994

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