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Properties

Label 2-1512-63.5-c1-0-15
Degree $2$
Conductor $1512$
Sign $0.354 + 0.935i$
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  + (−2.05 − 3.56i)5-s + (1.89 − 1.85i)7-s + (5.04 + 2.91i)11-s + (2.52 + 1.45i)13-s + (1.58 + 2.73i)17-s + (0.722 + 0.417i)19-s + (6.14 − 3.54i)23-s + (−5.96 + 10.3i)25-s + (1.91 − 1.10i)29-s − 4.23i·31-s + (−10.4 − 2.92i)35-s + (1.82 − 3.16i)37-s + (2.04 − 3.54i)41-s + (0.155 + 0.269i)43-s − 1.00·47-s + ⋯
L(s)  = 1  + (−0.920 − 1.59i)5-s + (0.714 − 0.699i)7-s + (1.52 + 0.877i)11-s + (0.699 + 0.403i)13-s + (0.383 + 0.664i)17-s + (0.165 + 0.0956i)19-s + (1.28 − 0.739i)23-s + (−1.19 + 2.06i)25-s + (0.355 − 0.205i)29-s − 0.760i·31-s + (−1.77 − 0.495i)35-s + (0.300 − 0.519i)37-s + (0.319 − 0.554i)41-s + (0.0237 + 0.0410i)43-s − 0.146·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.354 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.354 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.354 + 0.935i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (1097, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.354 + 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.801375080\)
\(L(\frac12)\) \(\approx\) \(1.801375080\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.89 + 1.85i)T \)
good5 \( 1 + (2.05 + 3.56i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.04 - 2.91i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.52 - 1.45i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.58 - 2.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.722 - 0.417i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.14 + 3.54i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.91 + 1.10i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.23iT - 31T^{2} \)
37 \( 1 + (-1.82 + 3.16i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.04 + 3.54i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.155 - 0.269i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.00T + 47T^{2} \)
53 \( 1 + (1.94 - 1.12i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 5.03T + 59T^{2} \)
61 \( 1 + 4.60iT - 61T^{2} \)
67 \( 1 + 9.99T + 67T^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 + (3.04 - 1.76i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 1.15T + 79T^{2} \)
83 \( 1 + (7.57 + 13.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.82 + 8.35i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.06 - 2.92i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.016828060898251104333404889652, −8.675311943679662742171590217449, −7.76832273133376843073605650927, −7.10666744777224120857798838815, −6.01276185523201545566033455105, −4.79582023820765722135942460380, −4.31380355062441507576446227729, −3.70933648558761703523174806628, −1.62288021009467560385207119636, −0.923567500133072496051584176302, 1.25853817712769071984801757458, 2.97521152729115475915350490653, 3.33205330247084467197278450595, 4.46071168285469174216926473568, 5.68110889395561612841956215302, 6.48934560355964939778901952859, 7.16858651108193848010236312655, 8.008270458533918068358626024106, 8.741557538046184578701243655831, 9.544135603898308253497473546631

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