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Properties

Label 2-93-93.26-c1-0-2
Degree $2$
Conductor $93$
Sign $-0.806 - 0.590i$
Analytic cond. $0.742608$
Root an. cond. $0.861747$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.52i·2-s + (0.5 + 1.65i)3-s − 4.37·4-s + (2.18 − 1.26i)5-s + (−4.18 + 1.26i)6-s + (1.18 − 2.05i)7-s − 5.98i·8-s + (−2.5 + 1.65i)9-s + (3.18 + 5.51i)10-s + (−2.18 − 3.78i)11-s + (−2.18 − 7.25i)12-s + (1.5 − 0.866i)13-s + (5.18 + 2.99i)14-s + (3.18 + 2.99i)15-s + 6.37·16-s + (−0.813 + 1.40i)17-s + ⋯
L(s)  = 1  + 1.78i·2-s + (0.288 + 0.957i)3-s − 2.18·4-s + (0.977 − 0.564i)5-s + (−1.70 + 0.515i)6-s + (0.448 − 0.776i)7-s − 2.11i·8-s + (−0.833 + 0.552i)9-s + (1.00 + 1.74i)10-s + (−0.659 − 1.14i)11-s + (−0.631 − 2.09i)12-s + (0.416 − 0.240i)13-s + (1.38 + 0.800i)14-s + (0.822 + 0.773i)15-s + 1.59·16-s + (−0.197 + 0.341i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.806 - 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.806 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $-0.806 - 0.590i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1/2),\ -0.806 - 0.590i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.332250 + 1.01639i\)
\(L(\frac12)\) \(\approx\) \(0.332250 + 1.01639i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 1.65i)T \)
31 \( 1 + (2 + 5.19i)T \)
good2 \( 1 - 2.52iT - 2T^{2} \)
5 \( 1 + (-2.18 + 1.26i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.18 + 2.05i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.18 + 3.78i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.5 + 0.866i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.813 - 1.40i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.87 - 6.70i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
37 \( 1 + (-1.5 - 0.866i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.18 + 1.26i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.5 + 4.33i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.63iT - 47T^{2} \)
53 \( 1 + (2.18 + 3.78i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.93 - 1.11i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + (-5.55 - 9.62i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.93 - 1.11i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.61 + 3.24i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.558 + 0.322i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.81 - 6.60i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 5.37T + 97T^{2} \)
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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

−14.51881241777169756253779195583, −13.84520423639732111396591366937, −13.12857739824922827184594688672, −10.78665273546725469492840377503, −9.800979783818281700045791703201, −8.558347267233582314970610155399, −8.028087802891129034071096298011, −6.12603654044057131353978305841, −5.36132705731939771445960864038, −4.07988246685019144117246273522, 1.97883922270571245807598700581, 2.70967460861016884886734706778, 4.95664019105079168711845381426, 6.68872737145980501703290502283, 8.505405764641327392799072030222, 9.446180202827691076626688677130, 10.57755569884181004104308425196, 11.57218087476733177393844940376, 12.54160085614027285515245826151, 13.29334644309284703491812412179

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