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Properties

Label 2-1512-63.59-c1-0-3
Degree $2$
Conductor $1512$
Sign $-0.308 - 0.951i$
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  + 2.84·5-s + (−2.64 + 0.0704i)7-s + 2.45i·11-s + (−3.06 − 1.76i)13-s + (−2.91 + 5.05i)17-s + (−2.90 + 1.67i)19-s + 8.01i·23-s + 3.10·25-s + (1.45 − 0.839i)29-s + (−3.45 + 1.99i)31-s + (−7.53 + 0.200i)35-s + (4.07 + 7.06i)37-s + (5.43 − 9.41i)41-s + (3.27 + 5.67i)43-s + (−3.31 + 5.74i)47-s + ⋯
L(s)  = 1  + 1.27·5-s + (−0.999 + 0.0266i)7-s + 0.738i·11-s + (−0.848 − 0.490i)13-s + (−0.708 + 1.22i)17-s + (−0.665 + 0.384i)19-s + 1.67i·23-s + 0.621·25-s + (0.269 − 0.155i)29-s + (−0.620 + 0.357i)31-s + (−1.27 + 0.0339i)35-s + (0.670 + 1.16i)37-s + (0.849 − 1.47i)41-s + (0.499 + 0.865i)43-s + (−0.483 + 0.837i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.308 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.308 - 0.951i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.308 - 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.194008474\)
\(L(\frac12)\) \(\approx\) \(1.194008474\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 - 0.0704i)T \)
good5 \( 1 - 2.84T + 5T^{2} \)
11 \( 1 - 2.45iT - 11T^{2} \)
13 \( 1 + (3.06 + 1.76i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.91 - 5.05i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.90 - 1.67i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 8.01iT - 23T^{2} \)
29 \( 1 + (-1.45 + 0.839i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.45 - 1.99i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.07 - 7.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.43 + 9.41i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.27 - 5.67i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.31 - 5.74i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.64 + 4.41i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.178 - 0.309i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.52 - 1.45i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.14 + 12.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.96iT - 71T^{2} \)
73 \( 1 + (-5.42 - 3.13i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.75 + 9.96i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.189 + 0.327i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.05 - 12.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.00 - 2.31i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.619920748357888835453279482680, −9.293707035753095569331465698689, −8.111715525252926508456355268347, −7.19427817972055361600171970024, −6.29341321194958620071304544240, −5.82052553631581161668892556739, −4.80444381140578970750324087052, −3.67680324944191291497401994746, −2.53166148699087978849594909110, −1.66486497629225605421768576769, 0.43055252211567319789582918915, 2.26595315877549038580853826916, 2.78660143424903609961521838172, 4.22293964045971202266904670355, 5.14340281158545426474958788216, 6.16860886705071344664837013534, 6.55929960482440630775951726095, 7.46724233312657409109526172881, 8.801309861794067319917329255104, 9.251170735434736427566033138405

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