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Properties

Label 2-1512-63.58-c1-0-16
Degree $2$
Conductor $1512$
Sign $0.562 + 0.826i$
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  + (0.927 + 1.60i)5-s + (0.900 − 2.48i)7-s + (−1.28 + 2.23i)11-s + (2.82 − 4.88i)13-s + (−3.57 − 6.19i)17-s + (0.636 − 1.10i)19-s + (0.120 + 0.208i)23-s + (0.777 − 1.34i)25-s + (−0.923 − 1.59i)29-s − 2.99·31-s + (4.83 − 0.862i)35-s + (0.338 − 0.585i)37-s + (0.733 − 1.27i)41-s + (4.14 + 7.17i)43-s + 12.3·47-s + ⋯
L(s)  = 1  + (0.414 + 0.718i)5-s + (0.340 − 0.940i)7-s + (−0.388 + 0.672i)11-s + (0.782 − 1.35i)13-s + (−0.868 − 1.50i)17-s + (0.146 − 0.252i)19-s + (0.0251 + 0.0435i)23-s + (0.155 − 0.269i)25-s + (−0.171 − 0.297i)29-s − 0.537·31-s + (0.817 − 0.145i)35-s + (0.0556 − 0.0963i)37-s + (0.114 − 0.198i)41-s + (0.631 + 1.09i)43-s + 1.79·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.562 + 0.826i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.562 + 0.826i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.725297120\)
\(L(\frac12)\) \(\approx\) \(1.725297120\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.900 + 2.48i)T \)
good5 \( 1 + (-0.927 - 1.60i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.28 - 2.23i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.82 + 4.88i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.57 + 6.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.636 + 1.10i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.120 - 0.208i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.923 + 1.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.99T + 31T^{2} \)
37 \( 1 + (-0.338 + 0.585i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.733 + 1.27i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.14 - 7.17i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 12.3T + 47T^{2} \)
53 \( 1 + (3.35 + 5.81i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 2.08T + 59T^{2} \)
61 \( 1 - 12.9T + 61T^{2} \)
67 \( 1 + 4.83T + 67T^{2} \)
71 \( 1 - 1.53T + 71T^{2} \)
73 \( 1 + (6.55 + 11.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 3.72T + 79T^{2} \)
83 \( 1 + (-3.00 - 5.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.60 - 11.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.40 - 11.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.523232617866017147991003409096, −8.496713589407675233254825437906, −7.52215767071114533790197893982, −7.11232574015473758457917034411, −6.13773366726196526382060734445, −5.17877123470684606125246912726, −4.31089473375300761899288962796, −3.16812080521340507709033547767, −2.28490752305747190831535226838, −0.71242718358527406794907632228, 1.43107675764921942962212345413, 2.27984261711735952098187742008, 3.71567750188127498639916958891, 4.59112377835654858649049571875, 5.68652415749752839637359427492, 6.03508955377225260533207693305, 7.15690744669542732377915064665, 8.366583752041791505232432698628, 8.818145786071714190515727607050, 9.234094018393049550124509470379

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