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Properties

Label 2-92-4.3-c4-0-38
Degree $2$
Conductor $92$
Sign $-0.146 + 0.989i$
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  + (3.98 + 0.292i)2-s − 14.7i·3-s + (15.8 + 2.33i)4-s + 5.38·5-s + (4.33 − 59.0i)6-s − 54.2i·7-s + (62.4 + 13.9i)8-s − 137.·9-s + (21.4 + 1.57i)10-s + 116. i·11-s + (34.5 − 234. i)12-s − 133.·13-s + (15.8 − 216. i)14-s − 79.6i·15-s + (245. + 73.9i)16-s + 296.·17-s + ⋯
L(s)  = 1  + (0.997 + 0.0732i)2-s − 1.64i·3-s + (0.989 + 0.146i)4-s + 0.215·5-s + (0.120 − 1.63i)6-s − 1.10i·7-s + (0.975 + 0.218i)8-s − 1.70·9-s + (0.214 + 0.0157i)10-s + 0.966i·11-s + (0.240 − 1.62i)12-s − 0.791·13-s + (0.0810 − 1.10i)14-s − 0.354i·15-s + (0.957 + 0.288i)16-s + 1.02·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92\)    =    \(2^{2} \cdot 23\)
Sign: $-0.146 + 0.989i$
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: \((2,\ 92,\ (\ :2),\ -0.146 + 0.989i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.05235 - 2.37751i\)
\(L(\frac12)\) \(\approx\) \(2.05235 - 2.37751i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.98 - 0.292i)T \)
23 \( 1 - 110. iT \)
good3 \( 1 + 14.7iT - 81T^{2} \)
5 \( 1 - 5.38T + 625T^{2} \)
7 \( 1 + 54.2iT - 2.40e3T^{2} \)
11 \( 1 - 116. iT - 1.46e4T^{2} \)
13 \( 1 + 133.T + 2.85e4T^{2} \)
17 \( 1 - 296.T + 8.35e4T^{2} \)
19 \( 1 + 395. iT - 1.30e5T^{2} \)
29 \( 1 - 1.23e3T + 7.07e5T^{2} \)
31 \( 1 - 647. iT - 9.23e5T^{2} \)
37 \( 1 - 804.T + 1.87e6T^{2} \)
41 \( 1 - 2.31e3T + 2.82e6T^{2} \)
43 \( 1 - 2.08e3iT - 3.41e6T^{2} \)
47 \( 1 - 2.72e3iT - 4.87e6T^{2} \)
53 \( 1 + 4.15e3T + 7.89e6T^{2} \)
59 \( 1 + 1.32e3iT - 1.21e7T^{2} \)
61 \( 1 - 3.37e3T + 1.38e7T^{2} \)
67 \( 1 - 3.93e3iT - 2.01e7T^{2} \)
71 \( 1 - 10.0iT - 2.54e7T^{2} \)
73 \( 1 + 4.27e3T + 2.83e7T^{2} \)
79 \( 1 + 9.48e3iT - 3.89e7T^{2} \)
83 \( 1 + 3.87e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.04e4T + 6.27e7T^{2} \)
97 \( 1 + 4.23e3T + 8.85e7T^{2} \)
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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

−12.99664668760812572458549835377, −12.39823126602685963856836599926, −11.38446819298813849706563688486, −9.985560429109670746462882056814, −7.74983489584819146422110024029, −7.24712760097288256069595627877, −6.23160911427369057507759341863, −4.65215151935053365559196243928, −2.68597999400733234376065400105, −1.22246879729988343765928606921, 2.70565379235235961485644551432, 3.92414501028168085036549745789, 5.29192166060623890041930540652, 5.94609635553247881333022069054, 8.161809854439384530501245944851, 9.585671919769432697230168774124, 10.39949243200252174858160934448, 11.57221204075652016830357849142, 12.37925481472763706513272360495, 14.00932107073689237939437346746

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