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Properties

Label 2-1218-203.86-c1-0-19
Degree $2$
Conductor $1218$
Sign $0.851 - 0.524i$
Analytic cond. $9.72577$
Root an. cond. $3.11861$
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  + (0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (0.110 − 0.191i)5-s − 0.999·6-s + (−1.52 − 2.16i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (0.191 − 0.110i)10-s + (−0.274 + 0.158i)11-s + (−0.866 − 0.499i)12-s − 1.21·13-s + (−0.242 − 2.63i)14-s + 0.220i·15-s + (−0.5 + 0.866i)16-s + (4.46 − 2.57i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (0.0494 − 0.0855i)5-s − 0.408·6-s + (−0.577 − 0.816i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (0.0605 − 0.0349i)10-s + (−0.0828 + 0.0478i)11-s + (−0.249 − 0.144i)12-s − 0.336·13-s + (−0.0647 − 0.704i)14-s + 0.0570i·15-s + (−0.125 + 0.216i)16-s + (1.08 − 0.624i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1218\)    =    \(2 \cdot 3 \cdot 7 \cdot 29\)
Sign: $0.851 - 0.524i$
Analytic conductor: \(9.72577\)
Root analytic conductor: \(3.11861\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1218} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1218,\ (\ :1/2),\ 0.851 - 0.524i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.959716099\)
\(L(\frac12)\) \(\approx\) \(1.959716099\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (1.52 + 2.16i)T \)
29 \( 1 + (-2.08 - 4.96i)T \)
good5 \( 1 + (-0.110 + 0.191i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.274 - 0.158i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.21T + 13T^{2} \)
17 \( 1 + (-4.46 + 2.57i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.73 - 3.89i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.31 + 2.27i)T + (-11.5 - 19.9i)T^{2} \)
31 \( 1 + (-7.64 + 4.41i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.27 + 1.88i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.58iT - 41T^{2} \)
43 \( 1 + 11.1iT - 43T^{2} \)
47 \( 1 + (-9.94 - 5.74i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.70 + 6.41i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.748 - 1.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.42 + 1.39i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.08 + 7.07i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.43T + 71T^{2} \)
73 \( 1 + (1.82 - 1.05i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.74 - 5.04i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.86T + 83T^{2} \)
89 \( 1 + (3.63 + 2.09i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.41iT - 97T^{2} \)
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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

−9.924510362613298250618207319534, −9.136179892838279832462764621944, −7.80529595629040828206320028024, −7.26750794679351657256379209259, −6.41117460440507870029147419153, −5.45784836188561447188975193116, −4.83678098306552590733810956353, −3.71236093211208611664773132087, −2.98579654702625488074762335518, −1.01344173818700447003396489994, 1.02918953389233171947086725043, 2.55255242217284009635016614786, 3.27723259416229207909034850107, 4.65459028341189877175383291205, 5.45037080928478007130618235857, 6.13583898516148144125634208088, 6.96586154254377490723219511393, 7.901307073839562159431601307899, 8.993548337175381901340410263217, 9.882197766118049603899293622710

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