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Properties

Label 2-114-57.8-c1-0-2
Degree $2$
Conductor $114$
Sign $0.999 + 0.0416i$
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  + (0.5 − 0.866i)2-s + (−0.724 + 1.57i)3-s + (−0.499 − 0.866i)4-s + (1.22 + 0.707i)5-s + (1 + 1.41i)6-s + 4.44·7-s − 0.999·8-s + (−1.94 − 2.28i)9-s + (1.22 − 0.707i)10-s + 0.317i·11-s + (1.72 − 0.158i)12-s + (−3 + 1.73i)13-s + (2.22 − 3.85i)14-s + (−2 + 1.41i)15-s + (−0.5 + 0.866i)16-s + (−5.44 − 3.14i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.418 + 0.908i)3-s + (−0.249 − 0.433i)4-s + (0.547 + 0.316i)5-s + (0.408 + 0.577i)6-s + 1.68·7-s − 0.353·8-s + (−0.649 − 0.760i)9-s + (0.387 − 0.223i)10-s + 0.0958i·11-s + (0.497 − 0.0458i)12-s + (−0.832 + 0.480i)13-s + (0.594 − 1.02i)14-s + (−0.516 + 0.365i)15-s + (−0.125 + 0.216i)16-s + (−1.32 − 0.763i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $0.999 + 0.0416i$
Analytic conductor: \(0.910294\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :1/2),\ 0.999 + 0.0416i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21218 - 0.0252802i\)
\(L(\frac12)\) \(\approx\) \(1.21218 - 0.0252802i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.724 - 1.57i)T \)
19 \( 1 + (4.17 - 1.25i)T \)
good5 \( 1 + (-1.22 - 0.707i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 4.44T + 7T^{2} \)
11 \( 1 - 0.317iT - 11T^{2} \)
13 \( 1 + (3 - 1.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.44 + 3.14i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-6.12 + 3.53i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.22 + 2.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.24iT - 31T^{2} \)
37 \( 1 + 0.778iT - 37T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.449 - 0.778i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.57 - 3.21i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.550 + 0.953i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.27 + 5.67i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.22 - 5.58i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.17 - 2.98i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.39 + 9.35i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.34 - 4.24i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.1iT - 83T^{2} \)
89 \( 1 + (-8.44 - 14.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (11.8 + 6.84i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.76468045180277074867120779802, −12.27869031735369679116517463167, −11.25128814210293305812136084988, −10.76314507542297780768563944811, −9.594401128101041311197933591779, −8.519337754733934367291991047061, −6.62789165325942128680394582565, −5.05717147512744661904457551374, −4.44535338726630202289195031464, −2.32452839037930250275437655529, 1.97480694420883125076863906035, 4.74228246569414955471926197973, 5.57999378999719840774515171077, 6.93738025068783551799881966503, 7.975051088910496573396710006444, 8.886082651476942872306757100776, 10.81319951215280699149901436075, 11.62113222233444643222066932306, 12.89596016263484521438243276991, 13.43996231120607674146444678181

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