Properties

Label 2-114-19.6-c1-0-1
Degree $2$
Conductor $114$
Sign $0.625 + 0.780i$
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 + (−0.939 − 0.342i)3-s + (0.173 − 0.984i)4-s + (−0.386 − 2.19i)5-s + (0.939 − 0.342i)6-s + (1.32 − 2.29i)7-s + (0.500 + 0.866i)8-s + (0.766 + 0.642i)9-s + (1.70 + 1.43i)10-s + (−1.11 − 1.92i)11-s + (−0.499 + 0.866i)12-s + (4.97 − 1.80i)13-s + (0.460 + 2.61i)14-s + (−0.386 + 2.19i)15-s + (−0.939 − 0.342i)16-s + (−2.61 + 2.19i)17-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (−0.542 − 0.197i)3-s + (0.0868 − 0.492i)4-s + (−0.172 − 0.980i)5-s + (0.383 − 0.139i)6-s + (0.501 − 0.868i)7-s + (0.176 + 0.306i)8-s + (0.255 + 0.214i)9-s + (0.539 + 0.452i)10-s + (−0.335 − 0.581i)11-s + (−0.144 + 0.250i)12-s + (1.37 − 0.501i)13-s + (0.123 + 0.698i)14-s + (−0.0998 + 0.566i)15-s + (−0.234 − 0.0855i)16-s + (−0.633 + 0.531i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $0.625 + 0.780i$
Analytic conductor: \(0.910294\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :1/2),\ 0.625 + 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.616738 - 0.296087i\)
\(L(\frac12)\) \(\approx\) \(0.616738 - 0.296087i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (4.29 - 0.725i)T \)
good5 \( 1 + (0.386 + 2.19i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-1.32 + 2.29i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.11 + 1.92i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.97 + 1.80i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (2.61 - 2.19i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-0.386 + 2.19i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-3.68 - 3.09i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (5.15 - 8.93i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.30T + 37T^{2} \)
41 \( 1 + (-6.79 - 2.47i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.02 + 5.83i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-8.43 - 7.07i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-1.70 + 9.67i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-3.79 + 3.18i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.990 - 5.61i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-6.56 - 5.51i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-0.764 - 4.33i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-2.62 - 0.956i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (12.9 + 4.72i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-5.25 + 9.09i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.34 - 2.67i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (13.6 - 11.4i)T + (16.8 - 95.5i)T^{2} \)
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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

−13.33727745929214365908643886214, −12.54779402593295595848789618784, −10.94168389695961434441244198858, −10.61669772985990870544685078844, −8.746844981662541520339392163629, −8.220116333500533526157778227697, −6.79548786358507246482755998169, −5.57526400348734760616258660642, −4.26451998751676803319155536322, −1.07252988782449084960696443483, 2.36041807562599896617885482882, 4.18851273398678178561434094120, 5.95524784806923228768015085896, 7.15015053932424157979659338868, 8.515144223974060452777299174682, 9.599656408947890903683874370517, 11.01637843528941480780184363389, 11.18787392170613307961638103881, 12.39978881400824320963065303032, 13.61265983657851174268184257546

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