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Properties

Label 2-93-31.4-c1-0-1
Degree $2$
Conductor $93$
Sign $-0.293 - 0.955i$
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  + (0.806 + 2.48i)2-s + (0.309 − 0.951i)3-s + (−3.88 + 2.82i)4-s + 0.899·5-s + 2.60·6-s + (−1.22 + 0.892i)7-s + (−5.91 − 4.29i)8-s + (−0.809 − 0.587i)9-s + (0.725 + 2.23i)10-s + (4.97 − 3.61i)11-s + (1.48 + 4.56i)12-s + (0.226 − 0.696i)13-s + (−3.20 − 2.32i)14-s + (0.278 − 0.855i)15-s + (2.92 − 9.00i)16-s + (−6.12 − 4.44i)17-s + ⋯
L(s)  = 1  + (0.569 + 1.75i)2-s + (0.178 − 0.549i)3-s + (−1.94 + 1.41i)4-s + 0.402·5-s + 1.06·6-s + (−0.464 + 0.337i)7-s + (−2.09 − 1.51i)8-s + (−0.269 − 0.195i)9-s + (0.229 + 0.705i)10-s + (1.50 − 1.09i)11-s + (0.428 + 1.31i)12-s + (0.0627 − 0.193i)13-s + (−0.856 − 0.621i)14-s + (0.0717 − 0.220i)15-s + (0.731 − 2.25i)16-s + (−1.48 − 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.293 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.293 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $-0.293 - 0.955i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1/2),\ -0.293 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.739667 + 1.00109i\)
\(L(\frac12)\) \(\approx\) \(0.739667 + 1.00109i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (5.25 - 1.83i)T \)
good2 \( 1 + (-0.806 - 2.48i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 - 0.899T + 5T^{2} \)
7 \( 1 + (1.22 - 0.892i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (-4.97 + 3.61i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.226 + 0.696i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (6.12 + 4.44i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.774 - 2.38i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-4.52 - 3.29i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.33 - 4.10i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + 3.76T + 37T^{2} \)
41 \( 1 + (0.725 + 2.23i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (0.134 + 0.414i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-2.73 + 8.41i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.19 + 0.871i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.21 - 3.74i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 - 3.67T + 61T^{2} \)
67 \( 1 - 2.90T + 67T^{2} \)
71 \( 1 + (-10.4 - 7.56i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (5.93 - 4.31i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-6.63 - 4.82i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.27 + 3.92i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-5.68 + 4.13i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-3.07 + 2.23i)T + (29.9 - 92.2i)T^{2} \)
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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

−14.20429394982443426442903799023, −13.73106663182644144729957084436, −12.80859496237566771807085280398, −11.54344185401102202986769857422, −9.217279539013560882782028850369, −8.706040457286960813197158916553, −7.16680278624723035955575096322, −6.41589479868231841708599531404, −5.40035708970782675543865204663, −3.59881859559331496431242696562, 2.03678952466704101227949371186, 3.78608299066550871619548544553, 4.63190764573104365181592430974, 6.46780968234335135888422425594, 9.024791838929573107899509651087, 9.598932490689189173497494238208, 10.63406630663478562796233233059, 11.52541337882927326015578042691, 12.64955184220536664846745145339, 13.44971143291335556644678098126

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