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Properties

Label 2-93-31.25-c1-0-4
Degree $2$
Conductor $93$
Sign $0.993 + 0.110i$
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  + 2.21·2-s + (−0.5 − 0.866i)3-s + 2.90·4-s + (−1.79 + 3.11i)5-s + (−1.10 − 1.91i)6-s + (−1.76 − 3.05i)7-s + 2·8-s + (−0.499 + 0.866i)9-s + (−3.97 + 6.88i)10-s + (1.41 − 2.45i)11-s + (−1.45 − 2.51i)12-s + (−0.0483 + 0.0838i)13-s + (−3.90 − 6.76i)14-s + 3.59·15-s − 1.37·16-s + (2.52 + 4.37i)17-s + ⋯
L(s)  = 1  + 1.56·2-s + (−0.288 − 0.499i)3-s + 1.45·4-s + (−0.803 + 1.39i)5-s + (−0.451 − 0.782i)6-s + (−0.666 − 1.15i)7-s + 0.707·8-s + (−0.166 + 0.288i)9-s + (−1.25 + 2.17i)10-s + (0.427 − 0.740i)11-s + (−0.419 − 0.725i)12-s + (−0.0134 + 0.0232i)13-s + (−1.04 − 1.80i)14-s + 0.927·15-s − 0.344·16-s + (0.612 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.993 + 0.110i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1/2),\ 0.993 + 0.110i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66293 - 0.0919693i\)
\(L(\frac12)\) \(\approx\) \(1.66293 - 0.0919693i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (0.0666 + 5.56i)T \)
good2 \( 1 - 2.21T + 2T^{2} \)
5 \( 1 + (1.79 - 3.11i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.76 + 3.05i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.41 + 2.45i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.0483 - 0.0838i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.52 - 4.37i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.95 - 3.38i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.428T + 23T^{2} \)
29 \( 1 - 8.02T + 29T^{2} \)
37 \( 1 + (5.06 + 8.78i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.484 - 0.839i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.54 - 6.13i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 13.0T + 47T^{2} \)
53 \( 1 + (1.63 - 2.82i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.68 - 4.65i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 1.09T + 61T^{2} \)
67 \( 1 + (0.645 - 1.11i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.11 + 1.93i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.433 - 0.750i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.76 - 3.05i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.79 + 6.57i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 4.77T + 89T^{2} \)
97 \( 1 + 1.19T + 97T^{2} \)
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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

−14.13411029733041032067745131492, −13.09395920223246641785492636157, −12.06433052387547275820197155441, −11.17515486213055319670978351249, −10.29137997240018442076041959699, −7.80594232528337443608567909674, −6.75753136381045887058237913550, −6.03188998695063826893709127633, −3.98689353197586536915876477886, −3.21387885935363586920174337546, 3.20743018464235405487038164332, 4.71175097191283110055107101420, 5.23813866148222044367934050289, 6.73482359120205313700883995462, 8.640567156017336375617372163050, 9.641040478788614944270932307930, 11.69085711682197061904824049905, 12.10062382944045707155441564346, 12.75452111157958632104842393953, 13.98176427754839189518095877755

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