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Properties

Label 2-92-4.3-c4-0-9
Degree $2$
Conductor $92$
Sign $0.716 - 0.697i$
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  + (1.55 − 3.68i)2-s + 3.76i·3-s + (−11.1 − 11.4i)4-s − 11.8·5-s + (13.8 + 5.84i)6-s + 89.5i·7-s + (−59.5 + 23.3i)8-s + 66.8·9-s + (−18.3 + 43.4i)10-s + 34.3i·11-s + (43.1 − 42.0i)12-s + 49.4·13-s + (329. + 139. i)14-s − 44.3i·15-s + (−6.71 + 255. i)16-s − 9.58·17-s + ⋯
L(s)  = 1  + (0.388 − 0.921i)2-s + 0.418i·3-s + (−0.697 − 0.716i)4-s − 0.472·5-s + (0.385 + 0.162i)6-s + 1.82i·7-s + (−0.931 + 0.364i)8-s + 0.825·9-s + (−0.183 + 0.434i)10-s + 0.284i·11-s + (0.299 − 0.291i)12-s + 0.292·13-s + (1.68 + 0.710i)14-s − 0.197i·15-s + (−0.0262 + 0.999i)16-s − 0.0331·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92\)    =    \(2^{2} \cdot 23\)
Sign: $0.716 - 0.697i$
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: \((2,\ 92,\ (\ :2),\ 0.716 - 0.697i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.38229 + 0.561976i\)
\(L(\frac12)\) \(\approx\) \(1.38229 + 0.561976i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.55 + 3.68i)T \)
23 \( 1 - 110. iT \)
good3 \( 1 - 3.76iT - 81T^{2} \)
5 \( 1 + 11.8T + 625T^{2} \)
7 \( 1 - 89.5iT - 2.40e3T^{2} \)
11 \( 1 - 34.3iT - 1.46e4T^{2} \)
13 \( 1 - 49.4T + 2.85e4T^{2} \)
17 \( 1 + 9.58T + 8.35e4T^{2} \)
19 \( 1 - 448. iT - 1.30e5T^{2} \)
29 \( 1 - 569.T + 7.07e5T^{2} \)
31 \( 1 + 281. iT - 9.23e5T^{2} \)
37 \( 1 - 167.T + 1.87e6T^{2} \)
41 \( 1 + 1.75e3T + 2.82e6T^{2} \)
43 \( 1 - 2.15e3iT - 3.41e6T^{2} \)
47 \( 1 + 2.24e3iT - 4.87e6T^{2} \)
53 \( 1 + 2.32e3T + 7.89e6T^{2} \)
59 \( 1 - 329. iT - 1.21e7T^{2} \)
61 \( 1 - 3.99e3T + 1.38e7T^{2} \)
67 \( 1 - 3.64e3iT - 2.01e7T^{2} \)
71 \( 1 + 3.43e3iT - 2.54e7T^{2} \)
73 \( 1 - 6.32e3T + 2.83e7T^{2} \)
79 \( 1 + 1.13e4iT - 3.89e7T^{2} \)
83 \( 1 + 5.28e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.22e4T + 6.27e7T^{2} \)
97 \( 1 + 7.38e3T + 8.85e7T^{2} \)
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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

−13.14013752827740680647263942506, −12.20748463856740580281055618366, −11.60861814178588278945037495182, −10.21966598100658418493551238613, −9.343508990215072022269556499890, −8.212521409064410822293119785598, −6.08016528108894758712507395481, −4.87860383377419209348547008052, −3.51658142924585274914538623060, −1.91091035229947916743605324383, 0.65606131887007219960037867448, 3.69624885498032537989631364376, 4.67476650450103173059396614530, 6.62747244813168866570070465431, 7.28109368351278153505034730643, 8.246720743384209231745571985529, 9.846038556135204490670202486760, 11.12451209558985342697856687018, 12.53251400496547791034975057792, 13.51450484820021155913276277100

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