Properties

Label 2-114-57.41-c1-0-3
Degree $2$
Conductor $114$
Sign $0.942 - 0.335i$
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  + (0.766 + 0.642i)2-s + (1.57 − 0.726i)3-s + (0.173 + 0.984i)4-s + (−1.96 − 0.346i)5-s + (1.67 + 0.453i)6-s + (0.910 + 1.57i)7-s + (−0.500 + 0.866i)8-s + (1.94 − 2.28i)9-s + (−1.28 − 1.52i)10-s + (−4.10 − 2.37i)11-s + (0.988 + 1.42i)12-s + (0.151 − 0.415i)13-s + (−0.316 + 1.79i)14-s + (−3.34 + 0.883i)15-s + (−0.939 + 0.342i)16-s + (−1.07 + 1.28i)17-s + ⋯
L(s)  = 1  + (0.541 + 0.454i)2-s + (0.907 − 0.419i)3-s + (0.0868 + 0.492i)4-s + (−0.879 − 0.155i)5-s + (0.682 + 0.185i)6-s + (0.344 + 0.596i)7-s + (−0.176 + 0.306i)8-s + (0.647 − 0.761i)9-s + (−0.405 − 0.483i)10-s + (−1.23 − 0.715i)11-s + (0.285 + 0.410i)12-s + (0.0419 − 0.115i)13-s + (−0.0845 + 0.479i)14-s + (−0.863 + 0.228i)15-s + (−0.234 + 0.0855i)16-s + (−0.260 + 0.310i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $0.942 - 0.335i$
Analytic conductor: \(0.910294\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :1/2),\ 0.942 - 0.335i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49621 + 0.258165i\)
\(L(\frac12)\) \(\approx\) \(1.49621 + 0.258165i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 - 0.642i)T \)
3 \( 1 + (-1.57 + 0.726i)T \)
19 \( 1 + (3.58 - 2.48i)T \)
good5 \( 1 + (1.96 + 0.346i)T + (4.69 + 1.71i)T^{2} \)
7 \( 1 + (-0.910 - 1.57i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.10 + 2.37i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.151 + 0.415i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (1.07 - 1.28i)T + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (-5.93 + 1.04i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (4.91 - 4.12i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-4.88 + 2.82i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.80iT - 37T^{2} \)
41 \( 1 + (-3.75 + 1.36i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-2.15 + 12.2i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-6.92 - 8.25i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (-0.424 - 2.40i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-3.87 - 3.24i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.80 - 10.2i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-5.27 - 6.28i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-0.897 + 5.08i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (13.5 - 4.94i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (3.23 + 8.88i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (0.523 - 0.302i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.07 - 1.48i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-1.64 + 1.96i)T + (-16.8 - 95.5i)T^{2} \)
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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

−13.58974974959484535616169150072, −12.83825116609298959338208667649, −11.92923526652684159073314435834, −10.65368682105266176963011858343, −8.789550721009369979572453258924, −8.193993943195314546999435137914, −7.23431799778637783113846043850, −5.68468573091249379506794748470, −4.12726707709151827256796262475, −2.71185842033480500845573931040, 2.55037531086282043472881374126, 4.00486293714043435025475025645, 4.91869151987006718148784347466, 7.16765443588319320242731379722, 8.016662358610621566038695409093, 9.413073016544847677072250199827, 10.58754978478886094180862602943, 11.28225463726360339545077380715, 12.78076046405316412276267741650, 13.44745848684962061472901996066

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