Properties

Label 2-114-57.53-c1-0-2
Degree $2$
Conductor $114$
Sign $0.742 + 0.669i$
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 + (0.779 − 1.54i)3-s + (0.766 + 0.642i)4-s + (1.86 + 2.21i)5-s + (−1.26 + 1.18i)6-s + (0.562 − 0.973i)7-s + (−0.500 − 0.866i)8-s + (−1.78 − 2.41i)9-s + (−0.990 − 2.72i)10-s + (2.70 − 1.56i)11-s + (1.59 − 0.683i)12-s + (−5.18 − 0.914i)13-s + (−0.861 + 0.722i)14-s + (4.88 − 1.14i)15-s + (0.173 + 0.984i)16-s + (−0.880 + 2.41i)17-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (0.450 − 0.892i)3-s + (0.383 + 0.321i)4-s + (0.832 + 0.992i)5-s + (−0.515 + 0.484i)6-s + (0.212 − 0.367i)7-s + (−0.176 − 0.306i)8-s + (−0.594 − 0.804i)9-s + (−0.313 − 0.860i)10-s + (0.816 − 0.471i)11-s + (0.459 − 0.197i)12-s + (−1.43 − 0.253i)13-s + (−0.230 + 0.193i)14-s + (1.26 − 0.296i)15-s + (0.0434 + 0.246i)16-s + (−0.213 + 0.586i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $0.742 + 0.669i$
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: \((2,\ 114,\ (\ :1/2),\ 0.742 + 0.669i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.915804 - 0.352065i\)
\(L(\frac12)\) \(\approx\) \(0.915804 - 0.352065i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.779 + 1.54i)T \)
19 \( 1 + (-4.13 - 1.37i)T \)
good5 \( 1 + (-1.86 - 2.21i)T + (-0.868 + 4.92i)T^{2} \)
7 \( 1 + (-0.562 + 0.973i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.70 + 1.56i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.18 + 0.914i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (0.880 - 2.41i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (4.31 - 5.14i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (1.09 - 0.399i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (3.90 + 2.25i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 12.0iT - 37T^{2} \)
41 \( 1 + (1.06 + 6.02i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-2.21 + 1.85i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-0.377 - 1.03i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (5.66 + 4.75i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-6.41 - 2.33i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (5.58 + 4.68i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (2.42 + 6.64i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (-3.31 + 2.77i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (1.30 + 7.41i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-4.30 + 0.759i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (12.5 + 7.22i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.38 - 13.5i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (1.47 - 4.04i)T + (-74.3 - 62.3i)T^{2} \)
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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

−13.66427826237649912143408123436, −12.31104568856506056439642418351, −11.39805036263220856863225636036, −10.13429726837942183447781668220, −9.330886072796559170468548003988, −7.891664228051453455907080235946, −7.04611530972122650699223196491, −5.96600367815899857367305574140, −3.28313176425333694518102268593, −1.85814406342591140448233541344, 2.23289135280257140548934229893, 4.58577735976762967335314219544, 5.57865271793607117840992155027, 7.33840282163774905826701155156, 8.765327301579738891238163984540, 9.395758070847481984996544770582, 10.02457024421410113496879280754, 11.52191309721304905826973403541, 12.60895502573802555619007871430, 14.11422858705964576081890397987

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