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Properties

Label 2-1512-8.5-c1-0-7
Degree $2$
Conductor $1512$
Sign $-0.0870 - 0.996i$
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  + (0.671 − 1.24i)2-s + (−1.09 − 1.67i)4-s + 3.66i·5-s + 7-s + (−2.81 + 0.246i)8-s + (4.56 + 2.46i)10-s + 4.45i·11-s − 1.51i·13-s + (0.671 − 1.24i)14-s + (−1.58 + 3.67i)16-s − 3.45·17-s − 2.29i·19-s + (6.12 − 4.02i)20-s + (5.54 + 2.99i)22-s − 8.76·23-s + ⋯
L(s)  = 1  + (0.474 − 0.880i)2-s + (−0.549 − 0.835i)4-s + 1.63i·5-s + 0.377·7-s + (−0.996 + 0.0870i)8-s + (1.44 + 0.778i)10-s + 1.34i·11-s − 0.420i·13-s + (0.179 − 0.332i)14-s + (−0.396 + 0.918i)16-s − 0.837·17-s − 0.527i·19-s + (1.37 − 0.901i)20-s + (1.18 + 0.637i)22-s − 1.82·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0870 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0870 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.0870 - 0.996i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.0870 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9746409414\)
\(L(\frac12)\) \(\approx\) \(0.9746409414\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.671 + 1.24i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 - 3.66iT - 5T^{2} \)
11 \( 1 - 4.45iT - 11T^{2} \)
13 \( 1 + 1.51iT - 13T^{2} \)
17 \( 1 + 3.45T + 17T^{2} \)
19 \( 1 + 2.29iT - 19T^{2} \)
23 \( 1 + 8.76T + 23T^{2} \)
29 \( 1 + 1.62iT - 29T^{2} \)
31 \( 1 + 9.26T + 31T^{2} \)
37 \( 1 - 5.69iT - 37T^{2} \)
41 \( 1 + 6.86T + 41T^{2} \)
43 \( 1 + 0.880iT - 43T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 - 11.5iT - 53T^{2} \)
59 \( 1 + 2.99iT - 59T^{2} \)
61 \( 1 - 8.51iT - 61T^{2} \)
67 \( 1 + 10.6iT - 67T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 - 7.85T + 73T^{2} \)
79 \( 1 - 4.57T + 79T^{2} \)
83 \( 1 - 2.60iT - 83T^{2} \)
89 \( 1 + 2.14T + 89T^{2} \)
97 \( 1 + 13.8T + 97T^{2} \)
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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

−9.940853953115479087531674802081, −9.248090253693427216494755624516, −8.013069593594494002716324551191, −7.12684811479885959795809753086, −6.42478534208656548508706581211, −5.47808662138445140344750029808, −4.40120206419842900348016400279, −3.65055739674971067309736992780, −2.51598804186948372762433360250, −1.94662188174026201275873644041, 0.30512160018162404754022879938, 1.93361679802607981246652262848, 3.72064911643734986960321603419, 4.24159439653686505811924177379, 5.36037344610666443774818794370, 5.67512609491827604801533756095, 6.72769336221139608958019463208, 7.86808538231793840464204373783, 8.418666053565695698245580785231, 8.910233497023251908867073400391

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