Properties

Label 2-1512-21.20-c1-0-6
Degree $2$
Conductor $1512$
Sign $-0.248 - 0.968i$
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  − 0.465·5-s + (0.656 + 2.56i)7-s + 5.28i·11-s − 4.50i·13-s + 3.15·17-s − 1.42i·19-s + 2.27i·23-s − 4.78·25-s + 6.09i·29-s + 2.76i·31-s + (−0.305 − 1.19i)35-s − 0.613·37-s − 6.93·41-s + 3.08·43-s − 9.30·47-s + ⋯
L(s)  = 1  − 0.208·5-s + (0.248 + 0.968i)7-s + 1.59i·11-s − 1.24i·13-s + 0.766·17-s − 0.327i·19-s + 0.474i·23-s − 0.956·25-s + 1.13i·29-s + 0.497i·31-s + (−0.0516 − 0.201i)35-s − 0.100·37-s − 1.08·41-s + 0.470·43-s − 1.35·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.248 - 0.968i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.248 - 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.289960452\)
\(L(\frac12)\) \(\approx\) \(1.289960452\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.656 - 2.56i)T \)
good5 \( 1 + 0.465T + 5T^{2} \)
11 \( 1 - 5.28iT - 11T^{2} \)
13 \( 1 + 4.50iT - 13T^{2} \)
17 \( 1 - 3.15T + 17T^{2} \)
19 \( 1 + 1.42iT - 19T^{2} \)
23 \( 1 - 2.27iT - 23T^{2} \)
29 \( 1 - 6.09iT - 29T^{2} \)
31 \( 1 - 2.76iT - 31T^{2} \)
37 \( 1 + 0.613T + 37T^{2} \)
41 \( 1 + 6.93T + 41T^{2} \)
43 \( 1 - 3.08T + 43T^{2} \)
47 \( 1 + 9.30T + 47T^{2} \)
53 \( 1 - 12.1iT - 53T^{2} \)
59 \( 1 + 3.62T + 59T^{2} \)
61 \( 1 + 2.27iT - 61T^{2} \)
67 \( 1 - 13.3T + 67T^{2} \)
71 \( 1 - 7.38iT - 71T^{2} \)
73 \( 1 - 10.8iT - 73T^{2} \)
79 \( 1 - 1.17T + 79T^{2} \)
83 \( 1 - 15.0T + 83T^{2} \)
89 \( 1 - 5.39T + 89T^{2} \)
97 \( 1 + 10.4iT - 97T^{2} \)
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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

−9.738415811428723733677164818939, −8.932542736255179668852095299233, −7.997452258778894664445260803651, −7.47223884845504120769087509779, −6.46971290224533396234313639039, −5.37143405303025820692970837480, −4.95575931457326070351373713537, −3.65649759943663318277596365898, −2.65279235663930172190285607596, −1.53662560176346796054151716198, 0.51592982276482041030238215057, 1.87297665175470809781489179145, 3.39935049420096903587766487248, 3.98594349992162772993507780028, 5.02221019865145908868156821853, 6.10533702725802727319690652010, 6.72742309394186127873569378831, 7.87799009427547933456313326132, 8.201706033488231428012041246923, 9.294823681243376470342062034943

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