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Properties

Label 2-1218-203.144-c1-0-36
Degree $2$
Conductor $1218$
Sign $-0.853 - 0.520i$
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 + (0.197 + 0.342i)5-s − 0.999·6-s + (−2.52 + 0.784i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (0.342 + 0.197i)10-s + (4.40 + 2.54i)11-s + (−0.866 + 0.499i)12-s − 4.89·13-s + (−1.79 + 1.94i)14-s − 0.395i·15-s + (−0.5 − 0.866i)16-s + (−5.96 − 3.44i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (0.0884 + 0.153i)5-s − 0.408·6-s + (−0.955 + 0.296i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.108 + 0.0625i)10-s + (1.32 + 0.767i)11-s + (−0.249 + 0.144i)12-s − 1.35·13-s + (−0.480 + 0.519i)14-s − 0.102i·15-s + (−0.125 − 0.216i)16-s + (−1.44 − 0.835i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.853 - 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1218\)    =    \(2 \cdot 3 \cdot 7 \cdot 29\)
Sign: $-0.853 - 0.520i$
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: \((2,\ 1218,\ (\ :1/2),\ -0.853 - 0.520i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.08194270082\)
\(L(\frac12)\) \(\approx\) \(0.08194270082\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.52 - 0.784i)T \)
29 \( 1 + (-1.39 - 5.20i)T \)
good5 \( 1 + (-0.197 - 0.342i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.40 - 2.54i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.89T + 13T^{2} \)
17 \( 1 + (5.96 + 3.44i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.56 - 2.05i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.50 + 7.80i)T + (-11.5 + 19.9i)T^{2} \)
31 \( 1 + (7.08 + 4.09i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (8.10 - 4.68i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.74iT - 41T^{2} \)
43 \( 1 - 9.05iT - 43T^{2} \)
47 \( 1 + (-6.01 + 3.47i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.23 - 7.33i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.790 + 1.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.02 + 0.593i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.06 + 1.85i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.0T + 71T^{2} \)
73 \( 1 + (-9.16 - 5.29i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.62 + 0.936i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.72T + 83T^{2} \)
89 \( 1 + (3.19 - 1.84i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.0iT - 97T^{2} \)
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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

−9.479489956924565344420244962518, −8.602961703373822034815752737755, −7.07144078830496097011692358263, −6.70613795514158174421768420809, −6.03285861119793110473346384486, −4.73001156433570645558791279406, −4.23406738946072187975145982147, −2.77708271103361932964172763137, −1.95476854067895879478093180932, −0.02737356664235244937613150455, 2.03554866995551954119266176679, 3.60447526345455261209710814348, 4.04928681788058117913820925364, 5.22707127854734831163593153645, 6.02935515377752146971847847451, 6.77610596476428376723736022441, 7.35085133280931109845896991768, 8.857800085590323583270043685120, 9.207006707723881653425671222623, 10.29416377751282948641710661789

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