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Properties

Label 2-92-4.3-c4-0-36
Degree $2$
Conductor $92$
Sign $-0.919 + 0.393i$
Analytic cond. $9.51003$
Root an. cond. $3.08383$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.20 + 3.33i)2-s − 10.3i·3-s + (−6.29 − 14.7i)4-s − 0.915·5-s + (34.5 + 22.7i)6-s − 20.9i·7-s + (62.9 + 11.3i)8-s − 26.1·9-s + (2.01 − 3.05i)10-s + 3.82i·11-s + (−152. + 65.1i)12-s − 248.·13-s + (69.8 + 46.0i)14-s + 9.47i·15-s + (−176. + 185. i)16-s − 321.·17-s + ⋯
L(s)  = 1  + (−0.550 + 0.834i)2-s − 1.14i·3-s + (−0.393 − 0.919i)4-s − 0.0366·5-s + (0.959 + 0.633i)6-s − 0.426i·7-s + (0.984 + 0.177i)8-s − 0.322·9-s + (0.0201 − 0.0305i)10-s + 0.0316i·11-s + (−1.05 + 0.452i)12-s − 1.47·13-s + (0.356 + 0.234i)14-s + 0.0420i·15-s + (−0.690 + 0.723i)16-s − 1.11·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 + 0.393i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.919 + 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92\)    =    \(2^{2} \cdot 23\)
Sign: $-0.919 + 0.393i$
Analytic conductor: \(9.51003\)
Root analytic conductor: \(3.08383\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{92} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 92,\ (\ :2),\ -0.919 + 0.393i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0791996 - 0.386310i\)
\(L(\frac12)\) \(\approx\) \(0.0791996 - 0.386310i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.20 - 3.33i)T \)
23 \( 1 + 110. iT \)
good3 \( 1 + 10.3iT - 81T^{2} \)
5 \( 1 + 0.915T + 625T^{2} \)
7 \( 1 + 20.9iT - 2.40e3T^{2} \)
11 \( 1 - 3.82iT - 1.46e4T^{2} \)
13 \( 1 + 248.T + 2.85e4T^{2} \)
17 \( 1 + 321.T + 8.35e4T^{2} \)
19 \( 1 - 80.0iT - 1.30e5T^{2} \)
29 \( 1 + 25.3T + 7.07e5T^{2} \)
31 \( 1 - 207. iT - 9.23e5T^{2} \)
37 \( 1 + 777.T + 1.87e6T^{2} \)
41 \( 1 + 669.T + 2.82e6T^{2} \)
43 \( 1 - 2.12e3iT - 3.41e6T^{2} \)
47 \( 1 - 447. iT - 4.87e6T^{2} \)
53 \( 1 + 878.T + 7.89e6T^{2} \)
59 \( 1 + 2.46e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.77e3T + 1.38e7T^{2} \)
67 \( 1 + 3.82e3iT - 2.01e7T^{2} \)
71 \( 1 + 5.34e3iT - 2.54e7T^{2} \)
73 \( 1 - 87.5T + 2.83e7T^{2} \)
79 \( 1 + 4.83e3iT - 3.89e7T^{2} \)
83 \( 1 + 1.15e4iT - 4.74e7T^{2} \)
89 \( 1 - 1.48e4T + 6.27e7T^{2} \)
97 \( 1 - 1.29e4T + 8.85e7T^{2} \)
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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

−13.10072073877689838593893913477, −11.94235000007149045752562367280, −10.48949574784973056299703016271, −9.359513145689202838929086453187, −7.964001586981492625029747577457, −7.22637654453969746849504576789, −6.29993750934432450029485352654, −4.68085155939505216908378941523, −1.92186093892922621455556408929, −0.21413898364096635063931085003, 2.35830931121882619097503553079, 3.94424148428477402228199023839, 5.06342392910027440398275887287, 7.26265492631731876605227710755, 8.745660664615616701898913008361, 9.605502829907799159057651616051, 10.38525281017054649366336832297, 11.46640435419394018114280578352, 12.41750865510392298285261557316, 13.65518734559243052902128843879

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