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Properties

Label 2-234-13.6-c2-0-0
Degree $2$
Conductor $234$
Sign $-0.654 + 0.755i$
Analytic cond. $6.37603$
Root an. cond. $2.52508$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 + 1.36i)2-s + (−1.73 + i)4-s + (−4.41 + 4.41i)5-s + (2.11 − 7.89i)7-s + (−2 − 1.99i)8-s + (−7.64 − 4.41i)10-s + (−18.5 + 4.96i)11-s + (−12.9 + 1.42i)13-s + 11.5·14-s + (1.99 − 3.46i)16-s + (19.9 − 11.5i)17-s + (0.417 + 0.111i)19-s + (3.23 − 12.0i)20-s + (−13.5 − 23.4i)22-s + (−36.4 − 21.0i)23-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.433 + 0.250i)4-s + (−0.882 + 0.882i)5-s + (0.302 − 1.12i)7-s + (−0.250 − 0.249i)8-s + (−0.764 − 0.441i)10-s + (−1.68 + 0.451i)11-s + (−0.993 + 0.109i)13-s + 0.825·14-s + (0.124 − 0.216i)16-s + (1.17 − 0.679i)17-s + (0.0219 + 0.00588i)19-s + (0.161 − 0.602i)20-s + (−0.616 − 1.06i)22-s + (−1.58 − 0.915i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.654 + 0.755i$
Analytic conductor: \(6.37603\)
Root analytic conductor: \(2.52508\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1),\ -0.654 + 0.755i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0432916 - 0.0947865i\)
\(L(\frac12)\) \(\approx\) \(0.0432916 - 0.0947865i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 - 1.36i)T \)
3 \( 1 \)
13 \( 1 + (12.9 - 1.42i)T \)
good5 \( 1 + (4.41 - 4.41i)T - 25iT^{2} \)
7 \( 1 + (-2.11 + 7.89i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (18.5 - 4.96i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (-19.9 + 11.5i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-0.417 - 0.111i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (36.4 + 21.0i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-3.25 + 5.64i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (17.8 - 17.8i)T - 961iT^{2} \)
37 \( 1 + (1.37 - 0.369i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (-10.8 - 40.3i)T + (-1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (51.1 - 29.5i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (15.0 + 15.0i)T + 2.20e3iT^{2} \)
53 \( 1 + 8.90T + 2.80e3T^{2} \)
59 \( 1 + (11.4 - 42.6i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-44.8 - 77.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-10.2 - 38.2i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (8.56 + 2.29i)T + (4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (5.92 + 5.92i)T + 5.32e3iT^{2} \)
79 \( 1 - 115.T + 6.24e3T^{2} \)
83 \( 1 + (-34.2 + 34.2i)T - 6.88e3iT^{2} \)
89 \( 1 + (58.0 - 15.5i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (129. + 34.6i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−12.59552718102631492752088146309, −11.65405922331961720003561813989, −10.44584463804462958791271403489, −9.957371023390319990924409093169, −7.979680808314934059712766505872, −7.64891326104398015341952563862, −6.84383655296221502850524157432, −5.24796412773882211001969438364, −4.21599018506570631844014197412, −2.88204582195715461706363044044, 0.04982759118664190443963266800, 2.16708264167266790440724954009, 3.59773109009902639756617329749, 5.07350987667052339845050116569, 5.60336474071645936206175529108, 7.891941531330935388493415698383, 8.197464634169711554571204928722, 9.485126556146236897404720608359, 10.45084754408385940588995244209, 11.62279984593902688107047093861

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