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Properties

Label 2-1218-203.144-c1-0-0
Degree $2$
Conductor $1218$
Sign $-0.873 - 0.487i$
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 + (−1.79 − 3.10i)5-s − 0.999·6-s + (−1.03 − 2.43i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (3.10 + 1.79i)10-s + (−0.742 − 0.428i)11-s + (0.866 − 0.499i)12-s − 6.04·13-s + (2.11 + 1.58i)14-s − 3.58i·15-s + (−0.5 − 0.866i)16-s + (1.53 + 0.885i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.800 − 1.38i)5-s − 0.408·6-s + (−0.392 − 0.919i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (0.980 + 0.566i)10-s + (−0.223 − 0.129i)11-s + (0.249 − 0.144i)12-s − 1.67·13-s + (0.565 + 0.424i)14-s − 0.924i·15-s + (−0.125 − 0.216i)16-s + (0.371 + 0.214i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 - 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.873 - 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1218\)    =    \(2 \cdot 3 \cdot 7 \cdot 29\)
Sign: $-0.873 - 0.487i$
Analytic conductor: \(9.72577\)
Root analytic conductor: \(3.11861\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1218} (1159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1218,\ (\ :1/2),\ -0.873 - 0.487i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.04147995763\)
\(L(\frac12)\) \(\approx\) \(0.04147995763\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.03 + 2.43i)T \)
29 \( 1 + (-5.27 + 1.06i)T \)
good5 \( 1 + (1.79 + 3.10i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.742 + 0.428i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.04T + 13T^{2} \)
17 \( 1 + (-1.53 - 0.885i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.39 - 1.38i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.91 - 5.05i)T + (-11.5 + 19.9i)T^{2} \)
31 \( 1 + (6.60 + 3.81i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.36 + 1.36i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.62iT - 41T^{2} \)
43 \( 1 - 12.2iT - 43T^{2} \)
47 \( 1 + (-9.57 + 5.53i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.38 - 4.13i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.23 - 5.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.28 - 3.62i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.92 + 3.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.5T + 71T^{2} \)
73 \( 1 + (3.26 + 1.88i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.26 + 2.46i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.75T + 83T^{2} \)
89 \( 1 + (-7.35 + 4.24i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.65iT - 97T^{2} \)
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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

−9.728987907862621587007755202354, −9.326699547085195000151945979053, −8.373842390829124827594359154801, −7.66078216855139946810400848063, −7.28861462949736985410567779273, −5.86471446978073902004778225792, −4.76378878706191832841760950816, −4.23928960386977501414043318339, −2.95952184971961529604325906367, −1.31584392143391305696455743925, 0.02069623857524929241943157098, 2.39139519370011405551626166950, 2.70186937114518748712201818048, 3.72113359266963505608302408150, 5.06687030705446220310302770413, 6.48479264063910833312862600277, 7.14114701200944357178279112595, 7.64727462583370074418965819138, 8.673188771057523277953088697459, 9.299393933041410248372271329966

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