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Properties

Label 2-1218-203.144-c1-0-17
Degree $2$
Conductor $1218$
Sign $0.898 + 0.439i$
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 + (−1.09 − 1.90i)5-s − 0.999·6-s + (−2.59 + 0.496i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (1.90 + 1.09i)10-s + (4.17 + 2.40i)11-s + (0.866 − 0.499i)12-s + 0.594·13-s + (2.00 − 1.72i)14-s − 2.19i·15-s + (−0.5 − 0.866i)16-s + (−6.33 − 3.65i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.491 − 0.851i)5-s − 0.408·6-s + (−0.982 + 0.187i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (0.602 + 0.347i)10-s + (1.25 + 0.725i)11-s + (0.249 − 0.144i)12-s + 0.164·13-s + (0.535 − 0.462i)14-s − 0.567i·15-s + (−0.125 − 0.216i)16-s + (−1.53 − 0.887i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 + 0.439i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.898 + 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1218\)    =    \(2 \cdot 3 \cdot 7 \cdot 29\)
Sign: $0.898 + 0.439i$
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: \((2,\ 1218,\ (\ :1/2),\ 0.898 + 0.439i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.149785568\)
\(L(\frac12)\) \(\approx\) \(1.149785568\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 - 0.496i)T \)
29 \( 1 + (2.42 + 4.80i)T \)
good5 \( 1 + (1.09 + 1.90i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.17 - 2.40i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.594T + 13T^{2} \)
17 \( 1 + (6.33 + 3.65i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.49 + 3.16i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.91 - 3.30i)T + (-11.5 + 19.9i)T^{2} \)
31 \( 1 + (-4.91 - 2.84i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.45 - 2.57i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.19iT - 41T^{2} \)
43 \( 1 + 0.843iT - 43T^{2} \)
47 \( 1 + (-7.67 + 4.42i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.10 - 1.91i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.27 + 7.40i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-12.8 + 7.42i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.60 + 9.71i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.95T + 71T^{2} \)
73 \( 1 + (-13.8 - 8.00i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.18 - 0.685i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.2T + 83T^{2} \)
89 \( 1 + (-2.58 + 1.49i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.38iT - 97T^{2} \)
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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

−9.306110673045574194594425247664, −9.094638871367466539976264173261, −8.275409044986054760103128053196, −7.03253731371110064638373807787, −6.81022874610496725294614551059, −5.38996112161338293191716392901, −4.49855422290098523557006341800, −3.55227291514770540040751519425, −2.24500746808853060665290676566, −0.68157652857861955586487008529, 1.12712005490651949023364132751, 2.62347855748205496530698569432, 3.47814891811642309475885640075, 4.05808644085750180159641743768, 6.01515132498547879280453035034, 6.74655574119656030435801712301, 7.23316900426854749564018428588, 8.361372107547120365116027843997, 8.951698917613898998881972519258, 9.705175623607737465864035047476

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