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Properties

Label 2-1512-63.16-c1-0-17
Degree $2$
Conductor $1512$
Sign $-0.553 + 0.833i$
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  + 0.468·5-s + (−2.39 + 1.13i)7-s + 1.34·11-s + (−3.16 − 5.48i)13-s + (2.47 + 4.28i)17-s + (2.38 − 4.13i)19-s − 7.62·23-s − 4.78·25-s + (1.80 − 3.12i)29-s + (−3.24 + 5.62i)31-s + (−1.11 + 0.531i)35-s + (5.24 − 9.07i)37-s + (0.0251 + 0.0435i)41-s + (−0.431 + 0.748i)43-s + (−5.49 − 9.51i)47-s + ⋯
L(s)  = 1  + 0.209·5-s + (−0.903 + 0.428i)7-s + 0.406·11-s + (−0.877 − 1.52i)13-s + (0.599 + 1.03i)17-s + (0.548 − 0.949i)19-s − 1.59·23-s − 0.956·25-s + (0.335 − 0.580i)29-s + (−0.583 + 1.01i)31-s + (−0.189 + 0.0897i)35-s + (0.861 − 1.49i)37-s + (0.00392 + 0.00680i)41-s + (−0.0658 + 0.114i)43-s + (−0.801 − 1.38i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.553 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.553 + 0.833i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.553 + 0.833i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7235896341\)
\(L(\frac12)\) \(\approx\) \(0.7235896341\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.39 - 1.13i)T \)
good5 \( 1 - 0.468T + 5T^{2} \)
11 \( 1 - 1.34T + 11T^{2} \)
13 \( 1 + (3.16 + 5.48i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.47 - 4.28i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.38 + 4.13i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.62T + 23T^{2} \)
29 \( 1 + (-1.80 + 3.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.24 - 5.62i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.24 + 9.07i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.0251 - 0.0435i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.431 - 0.748i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.49 + 9.51i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.84 + 10.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.93 - 3.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.87 + 3.24i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.32 + 2.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.04T + 71T^{2} \)
73 \( 1 + (3.30 + 5.71i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.58 + 2.75i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.90 - 8.49i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.30 - 9.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.97 + 12.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.425458288635191085564277730589, −8.312902739736675536562622307623, −7.69897150615365498604974682734, −6.68165871975461689908277856958, −5.84418602448399815789952549923, −5.28418359595481812174728767202, −3.92244944206733716562557163296, −3.09789340503506879316466353551, −2.03874240322681463692513878214, −0.27460223702929490993792920536, 1.52106256183489674713556451725, 2.75673056427501067040835857546, 3.84712228353312386298895715227, 4.59349131498431071176595252380, 5.83881792029445080186030272507, 6.46151230665881797433228298247, 7.35444872036210951247250229354, 7.974152953673921842358840051909, 9.400970095238395445699280729860, 9.578544373527729373297580870188

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