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Properties

Label 2-1512-63.16-c1-0-20
Degree $2$
Conductor $1512$
Sign $-0.422 + 0.906i$
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  − 0.526·5-s + (2.43 − 1.02i)7-s − 4.61·11-s + (0.244 + 0.423i)13-s + (−2.75 − 4.77i)17-s + (1.83 − 3.18i)19-s + 0.0539·23-s − 4.72·25-s + (3.28 − 5.68i)29-s + (−3.03 + 5.26i)31-s + (−1.28 + 0.538i)35-s + (0.223 − 0.387i)37-s + (−2.52 − 4.36i)41-s + (2.84 − 4.93i)43-s + (−4.59 − 7.96i)47-s + ⋯
L(s)  = 1  − 0.235·5-s + (0.922 − 0.386i)7-s − 1.39·11-s + (0.0678 + 0.117i)13-s + (−0.668 − 1.15i)17-s + (0.421 − 0.730i)19-s + 0.0112·23-s − 0.944·25-s + (0.609 − 1.05i)29-s + (−0.545 + 0.945i)31-s + (−0.216 + 0.0910i)35-s + (0.0367 − 0.0637i)37-s + (−0.394 − 0.682i)41-s + (0.434 − 0.752i)43-s + (−0.670 − 1.16i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.422 + 0.906i$
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: \((2,\ 1512,\ (\ :1/2),\ -0.422 + 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.019713014\)
\(L(\frac12)\) \(\approx\) \(1.019713014\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.43 + 1.02i)T \)
good5 \( 1 + 0.526T + 5T^{2} \)
11 \( 1 + 4.61T + 11T^{2} \)
13 \( 1 + (-0.244 - 0.423i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.75 + 4.77i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.83 + 3.18i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.0539T + 23T^{2} \)
29 \( 1 + (-3.28 + 5.68i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.03 - 5.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.223 + 0.387i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.52 + 4.36i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.84 + 4.93i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.59 + 7.96i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.37 - 7.57i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.31 - 5.74i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.232 - 0.403i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.59 - 4.49i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.76T + 71T^{2} \)
73 \( 1 + (5.23 + 9.07i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.18 + 14.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.49 - 7.78i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.05 + 12.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.22 + 9.04i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.091613776878333122885770966124, −8.403115654161467852715156002447, −7.47776369033302289566806297972, −7.14745455349020858778344681157, −5.76380023263860946407994994371, −4.97424134515801866189811204009, −4.31238637165609727551919784264, −3.00885606856853298113080472154, −2.02179872524003875820604720440, −0.38797655614470659273626257688, 1.57937850001283814262744614642, 2.61626757214501390306656699688, 3.81397201708231339574321862051, 4.80861162711176684656147092486, 5.54550150841364269517960482202, 6.37579982469976482496570015333, 7.69878127965195019614430168957, 7.985863092362498362920572495893, 8.750977819975905262742271098872, 9.800719619716761344908587958439

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