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Properties

Label 2-92-4.3-c4-0-39
Degree $2$
Conductor $92$
Sign $-0.979 + 0.202i$
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  + (2.52 − 3.10i)2-s − 12.4i·3-s + (−3.23 − 15.6i)4-s + 20.1·5-s + (−38.6 − 31.4i)6-s − 36.3i·7-s + (−56.7 − 29.5i)8-s − 73.9·9-s + (50.9 − 62.5i)10-s + 85.5i·11-s + (−195. + 40.2i)12-s + 308.·13-s + (−112. − 91.8i)14-s − 251. i·15-s + (−235. + 101. i)16-s − 305.·17-s + ⋯
L(s)  = 1  + (0.631 − 0.775i)2-s − 1.38i·3-s + (−0.202 − 0.979i)4-s + 0.806·5-s + (−1.07 − 0.873i)6-s − 0.742i·7-s + (−0.887 − 0.461i)8-s − 0.913·9-s + (0.509 − 0.625i)10-s + 0.707i·11-s + (−1.35 + 0.279i)12-s + 1.82·13-s + (−0.575 − 0.468i)14-s − 1.11i·15-s + (−0.918 + 0.396i)16-s − 1.05·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.202i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92\)    =    \(2^{2} \cdot 23\)
Sign: $-0.979 + 0.202i$
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: \((2,\ 92,\ (\ :2),\ -0.979 + 0.202i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.258164 - 2.52631i\)
\(L(\frac12)\) \(\approx\) \(0.258164 - 2.52631i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.52 + 3.10i)T \)
23 \( 1 + 110. iT \)
good3 \( 1 + 12.4iT - 81T^{2} \)
5 \( 1 - 20.1T + 625T^{2} \)
7 \( 1 + 36.3iT - 2.40e3T^{2} \)
11 \( 1 - 85.5iT - 1.46e4T^{2} \)
13 \( 1 - 308.T + 2.85e4T^{2} \)
17 \( 1 + 305.T + 8.35e4T^{2} \)
19 \( 1 - 565. iT - 1.30e5T^{2} \)
29 \( 1 - 1.14e3T + 7.07e5T^{2} \)
31 \( 1 + 974. iT - 9.23e5T^{2} \)
37 \( 1 - 905.T + 1.87e6T^{2} \)
41 \( 1 + 845.T + 2.82e6T^{2} \)
43 \( 1 + 372. iT - 3.41e6T^{2} \)
47 \( 1 - 287. iT - 4.87e6T^{2} \)
53 \( 1 - 2.02e3T + 7.89e6T^{2} \)
59 \( 1 + 6.49e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.96e3T + 1.38e7T^{2} \)
67 \( 1 - 1.56e3iT - 2.01e7T^{2} \)
71 \( 1 - 2.09e3iT - 2.54e7T^{2} \)
73 \( 1 - 257.T + 2.83e7T^{2} \)
79 \( 1 - 8.38e3iT - 3.89e7T^{2} \)
83 \( 1 - 5.12e3iT - 4.74e7T^{2} \)
89 \( 1 + 8.75e3T + 6.27e7T^{2} \)
97 \( 1 - 1.62e4T + 8.85e7T^{2} \)
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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

−13.06284685958894387418345084275, −12.02626470216955048285771599351, −10.86538290036252561547757373542, −9.832568428614136526144869209115, −8.282961292380900329197923878478, −6.68193562130927385205852588424, −5.94708697287877985479663908112, −4.04105283717787176386311730264, −2.10263779653695119573849941076, −1.10411101377099053317363725551, 2.99886358995618689399676895194, 4.39409267869817461818129180418, 5.56082471961373864314958853380, 6.47614479019614829863615688561, 8.702470441317923365861366239547, 9.036073723064408444756031234329, 10.63031295925142747290004832471, 11.59558726458795556732684047992, 13.29546017225696914201050520992, 13.80199737975976133405810309689

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