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Properties

Label 2-1512-24.11-c1-0-89
Degree $2$
Conductor $1512$
Sign $-0.140 + 0.990i$
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  + (1.25 − 0.648i)2-s + (1.15 − 1.63i)4-s + 1.27·5-s + i·7-s + (0.396 − 2.80i)8-s + (1.60 − 0.828i)10-s − 6.57i·11-s − 4.05i·13-s + (0.648 + 1.25i)14-s + (−1.31 − 3.77i)16-s + 6.09i·17-s − 2.08·19-s + (1.47 − 2.08i)20-s + (−4.26 − 8.26i)22-s + 6.85·23-s + ⋯
L(s)  = 1  + (0.888 − 0.458i)2-s + (0.578 − 0.815i)4-s + 0.570·5-s + 0.377i·7-s + (0.140 − 0.990i)8-s + (0.507 − 0.261i)10-s − 1.98i·11-s − 1.12i·13-s + (0.173 + 0.335i)14-s + (−0.329 − 0.944i)16-s + 1.47i·17-s − 0.477·19-s + (0.330 − 0.465i)20-s + (−0.909 − 1.76i)22-s + 1.42·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.140 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.140 + 0.990i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.140 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.101305036\)
\(L(\frac12)\) \(\approx\) \(3.101305036\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.25 + 0.648i)T \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 - 1.27T + 5T^{2} \)
11 \( 1 + 6.57iT - 11T^{2} \)
13 \( 1 + 4.05iT - 13T^{2} \)
17 \( 1 - 6.09iT - 17T^{2} \)
19 \( 1 + 2.08T + 19T^{2} \)
23 \( 1 - 6.85T + 23T^{2} \)
29 \( 1 + 6.53T + 29T^{2} \)
31 \( 1 - 3.26iT - 31T^{2} \)
37 \( 1 + 2.95iT - 37T^{2} \)
41 \( 1 + 3.35iT - 41T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 - 7.12T + 47T^{2} \)
53 \( 1 - 2.87T + 53T^{2} \)
59 \( 1 - 7.75iT - 59T^{2} \)
61 \( 1 + 12.0iT - 61T^{2} \)
67 \( 1 + 3.01T + 67T^{2} \)
71 \( 1 - 3.48T + 71T^{2} \)
73 \( 1 - 2.76T + 73T^{2} \)
79 \( 1 + 0.849iT - 79T^{2} \)
83 \( 1 - 15.8iT - 83T^{2} \)
89 \( 1 - 4.06iT - 89T^{2} \)
97 \( 1 + 4.90T + 97T^{2} \)
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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

−9.256689554036790250780487595349, −8.611116891270102652093165540180, −7.60959967288777870178732231722, −6.37312914573636420867225239068, −5.72432618920027305227449898615, −5.41330090239566688500031052957, −3.93392338759082727578717524981, −3.23423454625730399159728724260, −2.24870172712323811651541922794, −0.914624629995251700536274105643, 1.84094076520655931314008045454, 2.64151143068266269381271470083, 4.09449468248368047791825796428, 4.63171568605124497376738152327, 5.46398001687376882069981998918, 6.53050296360524275174858970442, 7.22661865485937512506689576980, 7.57219962464488532977809538059, 9.108286001058705730914973317352, 9.475889938739253922910265868404

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