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Properties

Label 2-1512-9.7-c1-0-11
Degree $2$
Conductor $1512$
Sign $-0.173 + 0.984i$
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  + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s + (−3 + 5.19i)11-s + (−3 − 5.19i)13-s + 2·17-s + 7·19-s + (−0.5 − 0.866i)23-s + (2 − 3.46i)25-s + (1 − 1.73i)29-s + (−5 − 8.66i)31-s + 0.999·35-s − 6·37-s + (−4 − 6.92i)41-s + (5 − 8.66i)43-s + (4 − 6.92i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (−0.188 + 0.327i)7-s + (−0.904 + 1.56i)11-s + (−0.832 − 1.44i)13-s + 0.485·17-s + 1.60·19-s + (−0.104 − 0.180i)23-s + (0.400 − 0.692i)25-s + (0.185 − 0.321i)29-s + (−0.898 − 1.55i)31-s + 0.169·35-s − 0.986·37-s + (−0.624 − 1.08i)41-s + (0.762 − 1.32i)43-s + (0.583 − 1.01i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9620103168\)
\(L(\frac12)\) \(\approx\) \(0.9620103168\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3 + 5.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1 + 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5 + 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + (4 + 6.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5 + 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6 - 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 15T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 4T + 89T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.356494441038606159857833832844, −8.378076043074565999211192672190, −7.44515635569073713841141414025, −7.26469473482514442409928978639, −5.56014337217688725155911928497, −5.31761957532217074093489959152, −4.25060008011484298563608053047, −3.02992069028306662202901952969, −2.13120076279300892756042564156, −0.39296884574117828429276621669, 1.33082789480518155517710834662, 2.97478546278579179579722839662, 3.42563124805557682894155882196, 4.80233870319004782314679167994, 5.51267915546044750538245619307, 6.56527448920911364196429455125, 7.32621864661580233909994686455, 7.954166626688268725006764886016, 9.018323949415207267546510017337, 9.595430475545117341480858550187

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