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Properties

Label 2-1512-63.59-c1-0-14
Degree $2$
Conductor $1512$
Sign $0.768 + 0.640i$
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  + 2.22·5-s + (−2.45 − 0.996i)7-s − 1.17i·11-s + (3.12 + 1.80i)13-s + (3.71 − 6.42i)17-s + (−3.05 + 1.76i)19-s + 5.81i·23-s − 0.0674·25-s + (6.04 − 3.48i)29-s + (6.88 − 3.97i)31-s + (−5.44 − 2.21i)35-s + (−5.54 − 9.60i)37-s + (0.809 − 1.40i)41-s + (0.904 + 1.56i)43-s + (4.26 − 7.38i)47-s + ⋯
L(s)  = 1  + 0.993·5-s + (−0.926 − 0.376i)7-s − 0.353i·11-s + (0.866 + 0.500i)13-s + (0.900 − 1.55i)17-s + (−0.700 + 0.404i)19-s + 1.21i·23-s − 0.0134·25-s + (1.12 − 0.648i)29-s + (1.23 − 0.713i)31-s + (−0.920 − 0.374i)35-s + (−0.911 − 1.57i)37-s + (0.126 − 0.218i)41-s + (0.137 + 0.238i)43-s + (0.622 − 1.07i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 + 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.768 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.768 + 0.640i$
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: \((2,\ 1512,\ (\ :1/2),\ 0.768 + 0.640i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.875723690\)
\(L(\frac12)\) \(\approx\) \(1.875723690\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.45 + 0.996i)T \)
good5 \( 1 - 2.22T + 5T^{2} \)
11 \( 1 + 1.17iT - 11T^{2} \)
13 \( 1 + (-3.12 - 1.80i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.71 + 6.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.05 - 1.76i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.81iT - 23T^{2} \)
29 \( 1 + (-6.04 + 3.48i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.88 + 3.97i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.54 + 9.60i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.809 + 1.40i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.904 - 1.56i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.26 + 7.38i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.62 - 5.55i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.00 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.09 + 4.09i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.96 + 8.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.67iT - 71T^{2} \)
73 \( 1 + (-6.92 - 3.99i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.25 + 3.91i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.390 - 0.677i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.75 + 3.03i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.49 - 2.01i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.480790657080336812549479744179, −8.806012206119636604639655781370, −7.70159796062419874120054356189, −6.85994821684249029690746484337, −6.03646476999355952847497382465, −5.53090192591038319854983770409, −4.21379162816505391705823689673, −3.29310074058936973481536883329, −2.24703680206654339041477309627, −0.843313221285970427875461485255, 1.25868330844478520013733897395, 2.50893141280859017985547521671, 3.39964179005909424050379141343, 4.56888144525200476027828159990, 5.68830663409043791634851266102, 6.27755002082023347745752175201, 6.81347101274385180637359526705, 8.343822665395034562473263235787, 8.596596936564712373119495366907, 9.714762568362523319277765257873

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