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Properties

Label 2-1512-63.59-c1-0-9
Degree $2$
Conductor $1512$
Sign $0.624 - 0.781i$
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  + 0.0525·5-s + (2.44 − 1.01i)7-s + 2.48i·11-s + (2.51 + 1.45i)13-s + (−2.88 + 4.99i)17-s + (−2.92 + 1.69i)19-s + 8.63i·23-s − 4.99·25-s + (6.23 − 3.60i)29-s + (8.59 − 4.96i)31-s + (0.128 − 0.0535i)35-s + (−0.770 − 1.33i)37-s + (−0.392 + 0.679i)41-s + (−2.03 − 3.51i)43-s + (−0.657 + 1.13i)47-s + ⋯
L(s)  = 1  + 0.0235·5-s + (0.922 − 0.385i)7-s + 0.749i·11-s + (0.698 + 0.403i)13-s + (−0.700 + 1.21i)17-s + (−0.671 + 0.387i)19-s + 1.79i·23-s − 0.999·25-s + (1.15 − 0.668i)29-s + (1.54 − 0.890i)31-s + (0.0216 − 0.00905i)35-s + (−0.126 − 0.219i)37-s + (−0.0612 + 0.106i)41-s + (−0.309 − 0.536i)43-s + (−0.0959 + 0.166i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.624 - 0.781i$
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: \((2,\ 1512,\ (\ :1/2),\ 0.624 - 0.781i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.785417518\)
\(L(\frac12)\) \(\approx\) \(1.785417518\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.44 + 1.01i)T \)
good5 \( 1 - 0.0525T + 5T^{2} \)
11 \( 1 - 2.48iT - 11T^{2} \)
13 \( 1 + (-2.51 - 1.45i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.88 - 4.99i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.92 - 1.69i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 8.63iT - 23T^{2} \)
29 \( 1 + (-6.23 + 3.60i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-8.59 + 4.96i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.770 + 1.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.392 - 0.679i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.03 + 3.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.657 - 1.13i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.710 + 0.410i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.32 - 4.01i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.87 - 2.81i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.95 - 12.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.51iT - 71T^{2} \)
73 \( 1 + (-6.75 - 3.90i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.50 - 12.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.14 - 5.44i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.93 + 13.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.57 + 0.909i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.774571379850158343456211732962, −8.522174980072655831044194173600, −8.161346259005274936750622265268, −7.21353776062176618276874340791, −6.33394900622989838263631949089, −5.50715901869709199078222147651, −4.29089164533761001647568371848, −3.95678128166154618327114763916, −2.26382715586038600022224365715, −1.38978812014999745863695484139, 0.76412522559425658897810419886, 2.23871266277443439844747709399, 3.15818806486029521134774220481, 4.52960951310911270661383977364, 5.02912563062716738529240405602, 6.22154418830859164963458251690, 6.75377970290048529454036593552, 8.102639040287217614835998973875, 8.431594114458336823219037129893, 9.153783132350922706746229532968

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