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Properties

Label 2-1512-63.16-c1-0-2
Degree $2$
Conductor $1512$
Sign $-0.000689 - 0.999i$
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  − 2.10·5-s + (−1.78 − 1.95i)7-s − 0.399·11-s + (1.44 + 2.49i)13-s + (0.176 + 0.305i)17-s + (2.84 − 4.93i)19-s + 0.877·23-s − 0.571·25-s + (−0.874 + 1.51i)29-s + (−4.56 + 7.91i)31-s + (3.75 + 4.11i)35-s + (−3.39 + 5.88i)37-s + (−1.20 − 2.08i)41-s + (0.276 − 0.479i)43-s + (5.86 + 10.1i)47-s + ⋯
L(s)  = 1  − 0.941·5-s + (−0.674 − 0.738i)7-s − 0.120·11-s + (0.400 + 0.693i)13-s + (0.0428 + 0.0741i)17-s + (0.653 − 1.13i)19-s + 0.182·23-s − 0.114·25-s + (−0.162 + 0.281i)29-s + (−0.820 + 1.42i)31-s + (0.634 + 0.694i)35-s + (−0.558 + 0.966i)37-s + (−0.187 − 0.325i)41-s + (0.0422 − 0.0730i)43-s + (0.856 + 1.48i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.000689 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.000689 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.000689 - 0.999i$
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: \((2,\ 1512,\ (\ :1/2),\ -0.000689 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7366532392\)
\(L(\frac12)\) \(\approx\) \(0.7366532392\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.78 + 1.95i)T \)
good5 \( 1 + 2.10T + 5T^{2} \)
11 \( 1 + 0.399T + 11T^{2} \)
13 \( 1 + (-1.44 - 2.49i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.176 - 0.305i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.84 + 4.93i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.877T + 23T^{2} \)
29 \( 1 + (0.874 - 1.51i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.56 - 7.91i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.39 - 5.88i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.20 + 2.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.276 + 0.479i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.86 - 10.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.07 - 3.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.66 - 8.07i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.03 - 8.72i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.601 - 1.04i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.6T + 71T^{2} \)
73 \( 1 + (-0.315 - 0.546i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.24 - 2.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.59 + 7.95i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.29 - 12.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.84 - 13.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.546722578709340596957605230094, −8.941461095975669530052737090235, −8.006240113597604488926580911701, −7.11749909812277868639323902059, −6.76996501894620712107480599661, −5.51561785909810843884337633136, −4.46687601394413846355615137539, −3.72927897454264052825347613019, −2.87588321833668689806244618229, −1.15607299260984939090717967226, 0.32982275125035272851674048255, 2.13077755143280790301926797277, 3.40697307089103635120764542607, 3.87639142246399079219155495461, 5.31217200143759950304047984969, 5.86449917446749195108081580203, 6.91957095338146372405676870338, 7.82772330068312149074228078666, 8.310558923815470796776910622642, 9.327561656155709582344809553939

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