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Properties

Label 2-93-93.92-c3-0-24
Degree $2$
Conductor $93$
Sign $0.433 + 0.901i$
Analytic cond. $5.48717$
Root an. cond. $2.34247$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.403i·2-s + (4.08 − 3.21i)3-s + 7.83·4-s − 16.8i·5-s + (1.29 + 1.64i)6-s − 11.2·7-s + 6.39i·8-s + (6.31 − 26.2i)9-s + 6.78·10-s − 45.0·11-s + (31.9 − 25.2i)12-s + 63.8i·13-s − 4.54i·14-s + (−54.0 − 68.5i)15-s + 60.1·16-s − 5.18·17-s + ⋯
L(s)  = 1  + 0.142i·2-s + (0.785 − 0.618i)3-s + 0.979·4-s − 1.50i·5-s + (0.0883 + 0.112i)6-s − 0.607·7-s + 0.282i·8-s + (0.233 − 0.972i)9-s + 0.214·10-s − 1.23·11-s + (0.769 − 0.606i)12-s + 1.36i·13-s − 0.0867i·14-s + (−0.930 − 1.18i)15-s + 0.939·16-s − 0.0739·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.433 + 0.901i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.433 + 0.901i$
Analytic conductor: \(5.48717\)
Root analytic conductor: \(2.34247\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :3/2),\ 0.433 + 0.901i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.85563 - 1.16646i\)
\(L(\frac12)\) \(\approx\) \(1.85563 - 1.16646i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.08 + 3.21i)T \)
31 \( 1 + (37.4 - 168. i)T \)
good2 \( 1 - 0.403iT - 8T^{2} \)
5 \( 1 + 16.8iT - 125T^{2} \)
7 \( 1 + 11.2T + 343T^{2} \)
11 \( 1 + 45.0T + 1.33e3T^{2} \)
13 \( 1 - 63.8iT - 2.19e3T^{2} \)
17 \( 1 + 5.18T + 4.91e3T^{2} \)
19 \( 1 - 138.T + 6.85e3T^{2} \)
23 \( 1 - 133.T + 1.21e4T^{2} \)
29 \( 1 - 137.T + 2.43e4T^{2} \)
37 \( 1 - 46.7iT - 5.06e4T^{2} \)
41 \( 1 - 262. iT - 6.89e4T^{2} \)
43 \( 1 + 79.0iT - 7.95e4T^{2} \)
47 \( 1 + 92.9iT - 1.03e5T^{2} \)
53 \( 1 + 408.T + 1.48e5T^{2} \)
59 \( 1 + 105. iT - 2.05e5T^{2} \)
61 \( 1 + 615. iT - 2.26e5T^{2} \)
67 \( 1 + 275.T + 3.00e5T^{2} \)
71 \( 1 - 405. iT - 3.57e5T^{2} \)
73 \( 1 - 781. iT - 3.89e5T^{2} \)
79 \( 1 + 1.21e3iT - 4.93e5T^{2} \)
83 \( 1 + 709.T + 5.71e5T^{2} \)
89 \( 1 + 1.46e3T + 7.04e5T^{2} \)
97 \( 1 - 780.T + 9.12e5T^{2} \)
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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

−13.20840332603125615293619967346, −12.48846816116249977482519227524, −11.54557423196122670721195090243, −9.794067607662003347232012707847, −8.792095283264341180900523181940, −7.71272246928639409161372347311, −6.64613315224814492785691770797, −5.07320964536350964812385109781, −3.00771712512271661384340641594, −1.36045551013067168653566687146, 2.83668095027724676397849449436, 3.10607097698076495873504254928, 5.59570677230403500290580914111, 7.13923357782527232923307974622, 7.85910368316217349603950299560, 9.788146838860126584005963039645, 10.48974238102034496297139233717, 11.16698185935562146577187478759, 12.79708174672585336699299205874, 13.87740181011913082403588180613

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