Properties

Label 2-114-57.53-c1-0-3
Degree $2$
Conductor $114$
Sign $-0.177 + 0.984i$
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  + (−0.939 − 0.342i)2-s + (−1.36 + 1.06i)3-s + (0.766 + 0.642i)4-s + (−2.20 − 2.62i)5-s + (1.64 − 0.532i)6-s + (1.68 − 2.92i)7-s + (−0.500 − 0.866i)8-s + (0.736 − 2.90i)9-s + (1.17 + 3.22i)10-s + (2.33 − 1.34i)11-s + (−1.73 − 0.0635i)12-s + (−5.05 − 0.891i)13-s + (−2.58 + 2.16i)14-s + (5.81 + 1.24i)15-s + (0.173 + 0.984i)16-s + (−1.44 + 3.97i)17-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (−0.789 + 0.614i)3-s + (0.383 + 0.321i)4-s + (−0.986 − 1.17i)5-s + (0.672 − 0.217i)6-s + (0.637 − 1.10i)7-s + (−0.176 − 0.306i)8-s + (0.245 − 0.969i)9-s + (0.371 + 1.01i)10-s + (0.704 − 0.406i)11-s + (−0.499 − 0.0183i)12-s + (−1.40 − 0.247i)13-s + (−0.690 + 0.579i)14-s + (1.50 + 0.321i)15-s + (0.0434 + 0.246i)16-s + (−0.350 + 0.963i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.177 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.177 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $-0.177 + 0.984i$
Analytic conductor: \(0.910294\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :1/2),\ -0.177 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.294748 - 0.352609i\)
\(L(\frac12)\) \(\approx\) \(0.294748 - 0.352609i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 + 0.342i)T \)
3 \( 1 + (1.36 - 1.06i)T \)
19 \( 1 + (2.73 + 3.39i)T \)
good5 \( 1 + (2.20 + 2.62i)T + (-0.868 + 4.92i)T^{2} \)
7 \( 1 + (-1.68 + 2.92i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.33 + 1.34i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.05 + 0.891i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (1.44 - 3.97i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-1.69 + 2.01i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-3.54 + 1.28i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-4.78 - 2.76i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.17iT - 37T^{2} \)
41 \( 1 + (-0.289 - 1.64i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-1.85 + 1.55i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-0.0440 - 0.120i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (6.53 + 5.48i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-3.87 - 1.41i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-3.53 - 2.96i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-3.81 - 10.4i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (-9.91 + 8.31i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (0.414 + 2.35i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-2.22 + 0.391i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (-6.27 - 3.62i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.209 + 1.18i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-3.13 + 8.60i)T + (-74.3 - 62.3i)T^{2} \)
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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

−12.84172918309906072161207244853, −12.02600054134056577491702268386, −11.16354826452743816736776987105, −10.29531263438741708060558475824, −9.031937518648027774524482526811, −8.051655494169910819200244201610, −6.78941436980547053160783146368, −4.82901439357249955812641348290, −4.05738755719554273715378554033, −0.70143382369172338886617560540, 2.38166520906636266129014018465, 4.86059034985705082045277190316, 6.43422377632880036121041126429, 7.25844847612783259186285276939, 8.172296915380548185226199411030, 9.712956491240172189896221341766, 10.98600511464253237888395100844, 11.78607992848047724784093925004, 12.22962924805537411482726122006, 14.22108432979124163853131494045

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