Properties

Label 2-114-57.29-c1-0-2
Degree $2$
Conductor $114$
Sign $-0.184 - 0.982i$
Analytic cond. $0.910294$
Root an. cond. $0.954093$
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  + (0.173 + 0.984i)2-s + (0.517 + 1.65i)3-s + (−0.939 + 0.342i)4-s + (−0.258 + 0.710i)5-s + (−1.53 + 0.797i)6-s + (0.777 − 1.34i)7-s + (−0.5 − 0.866i)8-s + (−2.46 + 1.71i)9-s + (−0.744 − 0.131i)10-s + (0.832 − 0.480i)11-s + (−1.05 − 1.37i)12-s + (0.416 − 0.496i)13-s + (1.46 + 0.532i)14-s + (−1.30 − 0.0594i)15-s + (0.766 − 0.642i)16-s + (6.73 − 1.18i)17-s + ⋯
L(s)  = 1  + (0.122 + 0.696i)2-s + (0.298 + 0.954i)3-s + (−0.469 + 0.171i)4-s + (−0.115 + 0.317i)5-s + (−0.627 + 0.325i)6-s + (0.294 − 0.509i)7-s + (−0.176 − 0.306i)8-s + (−0.821 + 0.570i)9-s + (−0.235 − 0.0415i)10-s + (0.250 − 0.144i)11-s + (−0.303 − 0.397i)12-s + (0.115 − 0.137i)13-s + (0.390 + 0.142i)14-s + (−0.337 − 0.0153i)15-s + (0.191 − 0.160i)16-s + (1.63 − 0.287i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $-0.184 - 0.982i$
Analytic conductor: \(0.910294\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :1/2),\ -0.184 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.729462 + 0.879312i\)
\(L(\frac12)\) \(\approx\) \(0.729462 + 0.879312i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 - 0.984i)T \)
3 \( 1 + (-0.517 - 1.65i)T \)
19 \( 1 + (4.14 + 1.35i)T \)
good5 \( 1 + (0.258 - 0.710i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (-0.777 + 1.34i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.832 + 0.480i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.416 + 0.496i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-6.73 + 1.18i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (-0.400 - 1.10i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (-1.39 + 7.92i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (2.63 + 1.52i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.12iT - 37T^{2} \)
41 \( 1 + (4.09 - 3.43i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (7.34 + 2.67i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (3.11 + 0.548i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (13.6 - 4.96i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-2.02 - 11.4i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-10.1 + 3.70i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (9.19 + 1.62i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-0.0322 - 0.0117i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (3.04 - 2.55i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (0.893 + 1.06i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (10.4 + 6.05i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.68 - 3.92i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-9.54 + 1.68i)T + (91.1 - 33.1i)T^{2} \)
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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

−14.25313797082788886889186664879, −13.19809185373099852823073772677, −11.67083358712823674657957585571, −10.57030123123334137049116453992, −9.610172311796264846283338001059, −8.416021669883106392291555840484, −7.41357765521036412336681041048, −5.87594332519679508753761771036, −4.58817094953926010843683071487, −3.34483724372996788864475836307, 1.61950410827600855092488691341, 3.31321013956548319010295501370, 5.16396058582476973217693566769, 6.57767324312862377559979415772, 8.120530073209348129558961756331, 8.837972739532904457400059254134, 10.22804778511427163997357574909, 11.56291898639289080631777791898, 12.38834027264263289499241841151, 12.94017169370280250183319474119

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