Properties

Label 2-114-57.53-c1-0-1
Degree $2$
Conductor $114$
Sign $0.913 - 0.406i$
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 + (−1.64 − 0.532i)3-s + (0.766 + 0.642i)4-s + (2.20 + 2.62i)5-s + (−1.36 − 1.06i)6-s + (1.68 − 2.92i)7-s + (0.500 + 0.866i)8-s + (2.43 + 1.75i)9-s + (1.17 + 3.22i)10-s + (−2.33 + 1.34i)11-s + (−0.920 − 1.46i)12-s + (−5.05 − 0.891i)13-s + (2.58 − 2.16i)14-s + (−2.23 − 5.50i)15-s + (0.173 + 0.984i)16-s + (1.44 − 3.97i)17-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (−0.951 − 0.307i)3-s + (0.383 + 0.321i)4-s + (0.986 + 1.17i)5-s + (−0.557 − 0.434i)6-s + (0.637 − 1.10i)7-s + (0.176 + 0.306i)8-s + (0.811 + 0.584i)9-s + (0.371 + 1.01i)10-s + (−0.704 + 0.406i)11-s + (−0.265 − 0.423i)12-s + (−1.40 − 0.247i)13-s + (0.690 − 0.579i)14-s + (−0.577 − 1.42i)15-s + (0.0434 + 0.246i)16-s + (0.350 − 0.963i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.406i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $0.913 - 0.406i$
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: \((2,\ 114,\ (\ :1/2),\ 0.913 - 0.406i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25975 + 0.267650i\)
\(L(\frac12)\) \(\approx\) \(1.25975 + 0.267650i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 + 0.532i)T \)
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

−13.73667765442207199295283927980, −12.78861320528338757512362309168, −11.55880834377094388268841203918, −10.59252396258458982936801431180, −9.994650215870714262556522696421, −7.34930368029207246876138271466, −7.13227752310428980090299907311, −5.66454342778512268519613821732, −4.64659707952525068874112988890, −2.42817775325617408845565512561, 1.99673939207776922023341824344, 4.58907747552965091142374312512, 5.40888061933744695064071710958, 6.11302743861923575838352111125, 8.203470988550847198434622855767, 9.557197144416556124973176135634, 10.42274904398465265804017554825, 11.80913384276449796760728130136, 12.42079264719493549154714938104, 13.17314182072920521149671237335

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