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Properties

Label 2-1512-63.59-c1-0-22
Degree $2$
Conductor $1512$
Sign $-0.856 + 0.516i$
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  + 0.203·5-s + (−1.27 − 2.32i)7-s − 4.46i·11-s + (1.25 + 0.725i)13-s + (1.60 − 2.78i)17-s + (−6.20 + 3.58i)19-s + 1.26i·23-s − 4.95·25-s + (0.944 − 0.545i)29-s + (−5.60 + 3.23i)31-s + (−0.258 − 0.471i)35-s + (3.02 + 5.24i)37-s + (0.370 − 0.642i)41-s + (−4.69 − 8.13i)43-s + (0.0465 − 0.0806i)47-s + ⋯
L(s)  = 1  + 0.0908·5-s + (−0.480 − 0.876i)7-s − 1.34i·11-s + (0.348 + 0.201i)13-s + (0.389 − 0.674i)17-s + (−1.42 + 0.821i)19-s + 0.264i·23-s − 0.991·25-s + (0.175 − 0.101i)29-s + (−1.00 + 0.580i)31-s + (−0.0436 − 0.0796i)35-s + (0.497 + 0.862i)37-s + (0.0578 − 0.100i)41-s + (−0.716 − 1.24i)43-s + (0.00679 − 0.0117i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.856 + 0.516i$
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: \((2,\ 1512,\ (\ :1/2),\ -0.856 + 0.516i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7580681222\)
\(L(\frac12)\) \(\approx\) \(0.7580681222\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.27 + 2.32i)T \)
good5 \( 1 - 0.203T + 5T^{2} \)
11 \( 1 + 4.46iT - 11T^{2} \)
13 \( 1 + (-1.25 - 0.725i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.60 + 2.78i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.20 - 3.58i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.26iT - 23T^{2} \)
29 \( 1 + (-0.944 + 0.545i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.60 - 3.23i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.02 - 5.24i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.370 + 0.642i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.69 + 8.13i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.0465 + 0.0806i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (9.35 + 5.39i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.16 + 8.94i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.34 + 4.24i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.02 - 6.97i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 15.6iT - 71T^{2} \)
73 \( 1 + (-0.984 - 0.568i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.86 + 10.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.29 - 3.98i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.52 - 6.10i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.17 - 1.83i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.167979840297698889152401559827, −8.290623295582905935289271629188, −7.63588918112301536446378840608, −6.53432747808654543508293944947, −6.06647912224187033660553298645, −4.96930087071419710556334044421, −3.80147811977985967724986412501, −3.27133153635025192918645733356, −1.73415321733524859293837290880, −0.28289690061603440779021982009, 1.80498566183471441033392977046, 2.65989585050819844260857226108, 3.92522638900433435604128895685, 4.76870550045735592848919351830, 5.86901978161587015646386687259, 6.41569329462235525325600856992, 7.42223286494731994953871965656, 8.243040571526936237810606112307, 9.130271206168558653454647057103, 9.666321967896529644558052103433

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