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Properties

Label 2-1512-21.17-c1-0-0
Degree $2$
Conductor $1512$
Sign $-0.866 + 0.500i$
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  + (1.59 + 2.76i)5-s + (−1.60 + 2.10i)7-s + (−4.75 − 2.74i)11-s − 6.96i·13-s + (−2.10 + 3.64i)17-s + (−5.72 + 3.30i)19-s + (−0.659 + 0.380i)23-s + (−2.60 + 4.50i)25-s − 4.20i·29-s + (−3.02 − 1.74i)31-s + (−8.38 − 1.06i)35-s + (−2.64 − 4.58i)37-s + 5.67·41-s − 3.39·43-s + (6.12 + 10.6i)47-s + ⋯
L(s)  = 1  + (0.714 + 1.23i)5-s + (−0.605 + 0.795i)7-s + (−1.43 − 0.827i)11-s − 1.93i·13-s + (−0.510 + 0.884i)17-s + (−1.31 + 0.758i)19-s + (−0.137 + 0.0794i)23-s + (−0.520 + 0.901i)25-s − 0.780i·29-s + (−0.543 − 0.313i)31-s + (−1.41 − 0.180i)35-s + (−0.435 − 0.753i)37-s + 0.886·41-s − 0.517·43-s + (0.893 + 1.54i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.866 + 0.500i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.866 + 0.500i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.07623234586\)
\(L(\frac12)\) \(\approx\) \(0.07623234586\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.60 - 2.10i)T \)
good5 \( 1 + (-1.59 - 2.76i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.75 + 2.74i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.96iT - 13T^{2} \)
17 \( 1 + (2.10 - 3.64i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.72 - 3.30i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.659 - 0.380i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.20iT - 29T^{2} \)
31 \( 1 + (3.02 + 1.74i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.64 + 4.58i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.67T + 41T^{2} \)
43 \( 1 + 3.39T + 43T^{2} \)
47 \( 1 + (-6.12 - 10.6i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.30 + 1.90i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.25 + 5.64i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.425 + 0.245i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.664 + 1.15i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.36iT - 71T^{2} \)
73 \( 1 + (11.8 + 6.83i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.27 + 7.39i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.45T + 83T^{2} \)
89 \( 1 + (0.817 + 1.41i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.6iT - 97T^{2} \)
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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

−10.29357522910365767942328705134, −9.192562714384277781917188083562, −8.183524135181669887451567452436, −7.72496949910118096179783532792, −6.35900936384737968052992898013, −5.96264630582859792247366852357, −5.37258437300785157002827383592, −3.70956392914083394800069121015, −2.79889925097634396543047613976, −2.28372189816626534816241061263, 0.02694204584582885286388485527, 1.65073786066383727837186435971, 2.57821796994137133835654429856, 4.25151082269265092052287033905, 4.67787514467316417434286013498, 5.55690738401942478225819710985, 6.83469971236010402021688060418, 7.10020899081849318206996754957, 8.454160648224551658752427408353, 9.073726371358555479112476646936

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