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Properties

Label 2-1218-7.2-c1-0-32
Degree $2$
Conductor $1218$
Sign $-0.983 + 0.182i$
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  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (1.09 − 1.90i)5-s − 0.999·6-s + (2.11 − 1.58i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−1.09 − 1.90i)10-s + (0.582 + 1.00i)11-s + (−0.499 + 0.866i)12-s − 2.19·13-s + (−0.317 − 2.62i)14-s − 2.19·15-s + (−0.5 + 0.866i)16-s + (−2.88 − 4.98i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.491 − 0.851i)5-s − 0.408·6-s + (0.799 − 0.600i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.347 − 0.602i)10-s + (0.175 + 0.304i)11-s + (−0.144 + 0.249i)12-s − 0.609·13-s + (−0.0849 − 0.701i)14-s − 0.567·15-s + (−0.125 + 0.216i)16-s + (−0.698 − 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1218\)    =    \(2 \cdot 3 \cdot 7 \cdot 29\)
Sign: $-0.983 + 0.182i$
Analytic conductor: \(9.72577\)
Root analytic conductor: \(3.11861\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1218} (1045, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1218,\ (\ :1/2),\ -0.983 + 0.182i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.788830512\)
\(L(\frac12)\) \(\approx\) \(1.788830512\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-2.11 + 1.58i)T \)
29 \( 1 - T \)
good5 \( 1 + (-1.09 + 1.90i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.582 - 1.00i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.19T + 13T^{2} \)
17 \( 1 + (2.88 + 4.98i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.23 + 3.87i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.234 + 0.406i)T + (-11.5 - 19.9i)T^{2} \)
31 \( 1 + (1.15 + 1.99i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.265 - 0.459i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.23T + 41T^{2} \)
43 \( 1 - 2.66T + 43T^{2} \)
47 \( 1 + (3.85 - 6.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.364 - 0.631i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.834 + 1.44i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.26 - 5.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.629 + 1.09i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 + (-3.40 - 5.89i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.28 - 3.96i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.02T + 83T^{2} \)
89 \( 1 + (-4.08 + 7.07i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.9T + 97T^{2} \)
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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

−9.381759998571256659754142240148, −8.744748574345235377893481189377, −7.56191479114448078027296898230, −6.94818729139165650484988057698, −5.71974915701314564976931518277, −4.84417797429420923035675435873, −4.48235641171045809054507266197, −2.82179010563050914685517795630, −1.74728100137564034360403150111, −0.70969645822147172818739097785, 1.95483445374374830800922415340, 3.15298713244096989646120139883, 4.22200519279119591412393636378, 5.17152507618965587315894907130, 5.93635804698959186135278782127, 6.56160181567005488570319171503, 7.60562441247606718138019464515, 8.445549367777358215774899044757, 9.213377684614387999340710864601, 10.21645868709903969737649668608

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