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Properties

Label 2-93-93.68-c1-0-2
Degree $2$
Conductor $93$
Sign $-0.639 - 0.768i$
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 + (1.68 + 0.396i)3-s − 4.37·4-s + (−2.18 − 1.26i)5-s + (−1 + 4.25i)6-s + (1.18 + 2.05i)7-s − 5.98i·8-s + (2.68 + 1.33i)9-s + (3.18 − 5.51i)10-s + (2.18 − 3.78i)11-s + (−7.37 − 1.73i)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 + (0.973 + 0.228i)3-s − 2.18·4-s + (−0.977 − 0.564i)5-s + (−0.408 + 1.73i)6-s + (0.448 + 0.776i)7-s − 2.11i·8-s + (0.895 + 0.445i)9-s + (1.00 − 1.74i)10-s + (0.659 − 1.14i)11-s + (−2.12 − 0.500i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 - 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $-0.639 - 0.768i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1/2),\ -0.639 - 0.768i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.461424 + 0.984172i\)
\(L(\frac12)\) \(\approx\) \(0.461424 + 0.984172i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.68 - 0.396i)T \)
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

−14.77988298775222196838322702785, −13.81828171719663473574396491129, −12.82328394773637947641799209273, −11.27155656904977364224498151639, −9.118122625051267789476107296279, −8.640538365061157208705804912197, −7.933543281024662936909554472508, −6.59004447091053942622807425829, −5.05002148452796651033938161443, −3.86086510677073623440021395319, 1.81240291444335917509984887674, 3.60377070637667245802667061325, 4.18717394366360591102584566641, 7.26826124412666833852912243837, 8.295421624743728262219990197537, 9.637302170051673650859005127972, 10.49776859577519645461867056123, 11.57269391475413636804719632330, 12.43681183801210723660961120512, 13.41430452341062027256104531499

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