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Properties

Label 2-92-4.3-c4-0-4
Degree $2$
Conductor $92$
Sign $0.00962 + 0.999i$
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  + (0.0192 + 3.99i)2-s + 15.0i·3-s + (−15.9 + 0.153i)4-s − 8.67·5-s + (−60.3 + 0.290i)6-s − 26.8i·7-s + (−0.923 − 63.9i)8-s − 146.·9-s + (−0.166 − 34.6i)10-s + 98.3i·11-s + (−2.32 − 241. i)12-s + 12.0·13-s + (107. − 0.516i)14-s − 130. i·15-s + (255. − 4.92i)16-s − 66.5·17-s + ⋯
L(s)  = 1  + (0.00481 + 0.999i)2-s + 1.67i·3-s + (−0.999 + 0.00962i)4-s − 0.346·5-s + (−1.67 + 0.00807i)6-s − 0.547i·7-s + (−0.0144 − 0.999i)8-s − 1.81·9-s + (−0.00166 − 0.346i)10-s + 0.812i·11-s + (−0.0161 − 1.67i)12-s + 0.0714·13-s + (0.547 − 0.00263i)14-s − 0.581i·15-s + (0.999 − 0.0192i)16-s − 0.230·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00962 + 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.00962 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92\)    =    \(2^{2} \cdot 23\)
Sign: $0.00962 + 0.999i$
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: \((2,\ 92,\ (\ :2),\ 0.00962 + 0.999i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.459873 - 0.455467i\)
\(L(\frac12)\) \(\approx\) \(0.459873 - 0.455467i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0192 - 3.99i)T \)
23 \( 1 + 110. iT \)
good3 \( 1 - 15.0iT - 81T^{2} \)
5 \( 1 + 8.67T + 625T^{2} \)
7 \( 1 + 26.8iT - 2.40e3T^{2} \)
11 \( 1 - 98.3iT - 1.46e4T^{2} \)
13 \( 1 - 12.0T + 2.85e4T^{2} \)
17 \( 1 + 66.5T + 8.35e4T^{2} \)
19 \( 1 - 385. iT - 1.30e5T^{2} \)
29 \( 1 - 200.T + 7.07e5T^{2} \)
31 \( 1 + 1.72e3iT - 9.23e5T^{2} \)
37 \( 1 + 2.03e3T + 1.87e6T^{2} \)
41 \( 1 + 2.01e3T + 2.82e6T^{2} \)
43 \( 1 - 539. iT - 3.41e6T^{2} \)
47 \( 1 - 1.91e3iT - 4.87e6T^{2} \)
53 \( 1 - 2.58e3T + 7.89e6T^{2} \)
59 \( 1 - 4.10e3iT - 1.21e7T^{2} \)
61 \( 1 + 4.28e3T + 1.38e7T^{2} \)
67 \( 1 - 8.14e3iT - 2.01e7T^{2} \)
71 \( 1 - 4.36e3iT - 2.54e7T^{2} \)
73 \( 1 + 2.12e3T + 2.83e7T^{2} \)
79 \( 1 + 2.99e3iT - 3.89e7T^{2} \)
83 \( 1 + 9.84e3iT - 4.74e7T^{2} \)
89 \( 1 - 5.76e3T + 6.27e7T^{2} \)
97 \( 1 + 9.92e3T + 8.85e7T^{2} \)
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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

−14.58624247263108051899734195688, −13.48449795777658937174097803775, −11.92563347330855202746909558575, −10.41147266267648498538429962767, −9.798548447099292937884605508609, −8.656871807613404943690051968431, −7.43718146964036533743983505708, −5.80708915800248737861286656544, −4.51531003536514612523929008552, −3.78401299318517742988139946456, 0.30804154107068181954212209363, 1.81050330124797630460438548801, 3.18685128865252965095877748501, 5.35515340852512154231169268153, 6.80348214568395421842443997604, 8.197454163608643052874643783512, 8.944122493647158119951244412929, 10.75892966136864695180659946738, 11.82570256771768301756446314261, 12.31264136979424246269526677007

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