Properties

Label 2-1512-24.11-c1-0-67
Degree $2$
Conductor $1512$
Sign $0.540 + 0.841i$
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  + (1.06 − 0.926i)2-s + (0.284 − 1.97i)4-s + 2.85·5-s + i·7-s + (−1.52 − 2.37i)8-s + (3.05 − 2.64i)10-s + 4.98i·11-s + 2.35i·13-s + (0.926 + 1.06i)14-s + (−3.83 − 1.12i)16-s − 8.19i·17-s + 5.90·19-s + (0.813 − 5.65i)20-s + (4.61 + 5.32i)22-s + 4.43·23-s + ⋯
L(s)  = 1  + (0.755 − 0.654i)2-s + (0.142 − 0.989i)4-s + 1.27·5-s + 0.377i·7-s + (−0.540 − 0.841i)8-s + (0.966 − 0.837i)10-s + 1.50i·11-s + 0.652i·13-s + (0.247 + 0.285i)14-s + (−0.959 − 0.281i)16-s − 1.98i·17-s + 1.35·19-s + (0.181 − 1.26i)20-s + (0.984 + 1.13i)22-s + 0.925·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.540 + 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.540 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.540 + 0.841i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.540 + 0.841i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.326931145\)
\(L(\frac12)\) \(\approx\) \(3.326931145\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.06 + 0.926i)T \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 - 2.85T + 5T^{2} \)
11 \( 1 - 4.98iT - 11T^{2} \)
13 \( 1 - 2.35iT - 13T^{2} \)
17 \( 1 + 8.19iT - 17T^{2} \)
19 \( 1 - 5.90T + 19T^{2} \)
23 \( 1 - 4.43T + 23T^{2} \)
29 \( 1 - 6.41T + 29T^{2} \)
31 \( 1 + 3.31iT - 31T^{2} \)
37 \( 1 + 5.51iT - 37T^{2} \)
41 \( 1 - 3.41iT - 41T^{2} \)
43 \( 1 - 1.37T + 43T^{2} \)
47 \( 1 - 2.60T + 47T^{2} \)
53 \( 1 + 13.3T + 53T^{2} \)
59 \( 1 - 10.6iT - 59T^{2} \)
61 \( 1 + 1.41iT - 61T^{2} \)
67 \( 1 - 0.221T + 67T^{2} \)
71 \( 1 + 0.398T + 71T^{2} \)
73 \( 1 + 10.7T + 73T^{2} \)
79 \( 1 + 9.76iT - 79T^{2} \)
83 \( 1 + 2.06iT - 83T^{2} \)
89 \( 1 - 17.6iT - 89T^{2} \)
97 \( 1 + 3.62T + 97T^{2} \)
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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

−9.497579359702637776955726551895, −9.165371006241543549702218748858, −7.39326944272496285160303280186, −6.80660794019320379016254295426, −5.83237600176372002043594856210, −5.03879945028469794064698063228, −4.53340472373168207957872996657, −2.96069892812514495890978048655, −2.32942376430752696959926470730, −1.27575501263964985491544603786, 1.36024638331037826456394177186, 2.90383251480823021871740385854, 3.52459311206368728110847747547, 4.83346049332819098573530899633, 5.68875652510476740883693752182, 6.10518657742167293689098599981, 6.91246965357239653025162282890, 8.078562088870693862707251776570, 8.531553107147852331238137613853, 9.520159799445252846693592707669

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