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Properties

Label 2-1512-24.11-c1-0-72
Degree $2$
Conductor $1512$
Sign $0.995 + 0.0904i$
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  + (−0.743 + 1.20i)2-s + (−0.893 − 1.78i)4-s + 3.69·5-s + i·7-s + (2.81 + 0.255i)8-s + (−2.74 + 4.44i)10-s − 2.05i·11-s − 4.65i·13-s + (−1.20 − 0.743i)14-s + (−2.40 + 3.19i)16-s − 5.73i·17-s + 3.27·19-s + (−3.30 − 6.61i)20-s + (2.47 + 1.53i)22-s + 4.45·23-s + ⋯
L(s)  = 1  + (−0.525 + 0.850i)2-s + (−0.446 − 0.894i)4-s + 1.65·5-s + 0.377i·7-s + (0.995 + 0.0904i)8-s + (−0.869 + 1.40i)10-s − 0.620i·11-s − 1.28i·13-s + (−0.321 − 0.198i)14-s + (−0.600 + 0.799i)16-s − 1.39i·17-s + 0.750·19-s + (−0.738 − 1.47i)20-s + (0.527 + 0.326i)22-s + 0.928·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.995 + 0.0904i$
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: \((2,\ 1512,\ (\ :1/2),\ 0.995 + 0.0904i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.640784529\)
\(L(\frac12)\) \(\approx\) \(1.640784529\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.743 - 1.20i)T \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 - 3.69T + 5T^{2} \)
11 \( 1 + 2.05iT - 11T^{2} \)
13 \( 1 + 4.65iT - 13T^{2} \)
17 \( 1 + 5.73iT - 17T^{2} \)
19 \( 1 - 3.27T + 19T^{2} \)
23 \( 1 - 4.45T + 23T^{2} \)
29 \( 1 + 10.2T + 29T^{2} \)
31 \( 1 - 1.26iT - 31T^{2} \)
37 \( 1 + 3.25iT - 37T^{2} \)
41 \( 1 + 8.72iT - 41T^{2} \)
43 \( 1 + 2.56T + 43T^{2} \)
47 \( 1 + 2.03T + 47T^{2} \)
53 \( 1 + 10.0T + 53T^{2} \)
59 \( 1 - 0.851iT - 59T^{2} \)
61 \( 1 - 2.31iT - 61T^{2} \)
67 \( 1 - 15.4T + 67T^{2} \)
71 \( 1 - 4.44T + 71T^{2} \)
73 \( 1 + 6.97T + 73T^{2} \)
79 \( 1 - 11.6iT - 79T^{2} \)
83 \( 1 + 12.8iT - 83T^{2} \)
89 \( 1 + 2.35iT - 89T^{2} \)
97 \( 1 - 4.02T + 97T^{2} \)
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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

−9.291336901604708587572879803611, −8.918454692519828818782074084732, −7.79370996627259371230760184719, −7.05011056052412157317628677041, −6.11120779538085146881808895869, −5.34498672242058475540433954165, −5.19361103275894119752425008620, −3.24892430893707496149623391375, −2.11303116943221631818103607037, −0.810335297758381467505132920582, 1.51902619989215324194990773305, 1.94233005293310454524324147974, 3.22100284443397321969866610846, 4.32460129042719915362844313676, 5.23177971349009285261086287267, 6.34184307231280267217180249443, 7.07085395563971998601148508949, 8.086352948733322579826046331845, 9.102946728755501984652159417486, 9.595904805777767836991092131715

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