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Properties

Label 2-1512-63.25-c1-0-4
Degree $2$
Conductor $1512$
Sign $-0.997 - 0.0709i$
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  + (−1.71 + 2.97i)5-s + (−0.727 + 2.54i)7-s + (2.20 + 3.81i)11-s + (1.49 + 2.58i)13-s + (−0.542 + 0.939i)17-s + (−3.74 − 6.48i)19-s + (−2.16 + 3.74i)23-s + (−3.40 − 5.89i)25-s + (−1.68 + 2.91i)29-s + 9.37·31-s + (−6.31 − 6.53i)35-s + (−2.50 − 4.34i)37-s + (1.20 + 2.08i)41-s + (3.31 − 5.74i)43-s − 3.00·47-s + ⋯
L(s)  = 1  + (−0.768 + 1.33i)5-s + (−0.275 + 0.961i)7-s + (0.664 + 1.15i)11-s + (0.414 + 0.717i)13-s + (−0.131 + 0.227i)17-s + (−0.858 − 1.48i)19-s + (−0.450 + 0.781i)23-s + (−0.680 − 1.17i)25-s + (−0.312 + 0.541i)29-s + 1.68·31-s + (−1.06 − 1.10i)35-s + (−0.412 − 0.714i)37-s + (0.187 + 0.325i)41-s + (0.505 − 0.875i)43-s − 0.438·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.997 - 0.0709i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (1369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.997 - 0.0709i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9548748304\)
\(L(\frac12)\) \(\approx\) \(0.9548748304\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.727 - 2.54i)T \)
good5 \( 1 + (1.71 - 2.97i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.20 - 3.81i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.49 - 2.58i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.542 - 0.939i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.74 + 6.48i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.16 - 3.74i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.68 - 2.91i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.37T + 31T^{2} \)
37 \( 1 + (2.50 + 4.34i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.20 - 2.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.31 + 5.74i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.00T + 47T^{2} \)
53 \( 1 + (-0.530 + 0.919i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 + 5.42T + 61T^{2} \)
67 \( 1 - 3.33T + 67T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + (8.21 - 14.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 2.35T + 79T^{2} \)
83 \( 1 + (-1.60 + 2.78i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.67 + 9.82i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.40 - 11.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.782065011328824385168296609059, −9.112407189278823401369460592380, −8.295169786773226573122326835996, −7.19637129426388887837878374516, −6.76489886319923311010516544627, −6.04648542255154570239535312883, −4.69157130250678338081838337558, −3.88973034518251535858094734317, −2.86589045986787481505150686027, −1.97742154527383585267739564590, 0.40362910942617551883922371347, 1.31776134689157671126381661577, 3.22456773875184255993623917041, 4.06455211360418446815058215693, 4.62767188373296120969509247882, 5.90680667526887086061017450599, 6.48615385763865862567208140182, 7.924014698372084314981778635088, 8.129098799241047445888895668086, 8.900926808540291179639618583367

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