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Properties

Label 2-93-31.15-c2-0-5
Degree $2$
Conductor $93$
Sign $0.846 + 0.531i$
Analytic cond. $2.53406$
Root an. cond. $1.59187$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.30 + 1.67i)2-s + (1.01 − 1.40i)3-s + (1.26 − 3.88i)4-s − 2.14·5-s + 4.92i·6-s + (2.44 − 7.51i)7-s + (0.0737 + 0.226i)8-s + (−0.927 − 2.85i)9-s + (4.92 − 3.57i)10-s + (3.14 + 1.02i)11-s + (−4.15 − 5.72i)12-s + (10.0 − 13.7i)13-s + (6.94 + 21.3i)14-s + (−2.18 + 3.00i)15-s + (12.6 + 9.20i)16-s + (8.18 − 2.65i)17-s + ⋯
L(s)  = 1  + (−1.15 + 0.835i)2-s + (0.339 − 0.467i)3-s + (0.315 − 0.971i)4-s − 0.428·5-s + 0.820i·6-s + (0.348 − 1.07i)7-s + (0.00921 + 0.0283i)8-s + (−0.103 − 0.317i)9-s + (0.492 − 0.357i)10-s + (0.285 + 0.0928i)11-s + (−0.346 − 0.476i)12-s + (0.770 − 1.06i)13-s + (0.496 + 1.52i)14-s + (−0.145 + 0.200i)15-s + (0.791 + 0.575i)16-s + (0.481 − 0.156i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.531i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.846 + 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.846 + 0.531i$
Analytic conductor: \(2.53406\)
Root analytic conductor: \(1.59187\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1),\ 0.846 + 0.531i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.714180 - 0.205697i\)
\(L(\frac12)\) \(\approx\) \(0.714180 - 0.205697i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.01 + 1.40i)T \)
31 \( 1 + (30.9 - 1.38i)T \)
good2 \( 1 + (2.30 - 1.67i)T + (1.23 - 3.80i)T^{2} \)
5 \( 1 + 2.14T + 25T^{2} \)
7 \( 1 + (-2.44 + 7.51i)T + (-39.6 - 28.8i)T^{2} \)
11 \( 1 + (-3.14 - 1.02i)T + (97.8 + 71.1i)T^{2} \)
13 \( 1 + (-10.0 + 13.7i)T + (-52.2 - 160. i)T^{2} \)
17 \( 1 + (-8.18 + 2.65i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-8.96 + 6.51i)T + (111. - 343. i)T^{2} \)
23 \( 1 + (-0.696 + 0.226i)T + (427. - 310. i)T^{2} \)
29 \( 1 + (22.4 + 30.8i)T + (-259. + 799. i)T^{2} \)
37 \( 1 - 6.54iT - 1.36e3T^{2} \)
41 \( 1 + (-13.1 + 9.53i)T + (519. - 1.59e3i)T^{2} \)
43 \( 1 + (-37.7 - 51.8i)T + (-571. + 1.75e3i)T^{2} \)
47 \( 1 + (-28.6 - 20.8i)T + (682. + 2.10e3i)T^{2} \)
53 \( 1 + (-49.3 + 16.0i)T + (2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (54.6 + 39.7i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 - 89.5iT - 3.72e3T^{2} \)
67 \( 1 + 61.5T + 4.48e3T^{2} \)
71 \( 1 + (-4.28 - 13.1i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (-99.4 - 32.3i)T + (4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (20.8 - 6.76i)T + (5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (19.1 + 26.3i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 + (-156. - 50.9i)T + (6.40e3 + 4.65e3i)T^{2} \)
97 \( 1 + (-11.2 + 34.7i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−13.84240841655198945650724044826, −12.79805172307000793921930819557, −11.30663699530911815290497120839, −10.16469209614685921818748837028, −9.013238450352948397435739738589, −7.74334840957228824641451085571, −7.49234381762282248593391207384, −5.99220150357630630158906305053, −3.76056886633055561302107466687, −0.885155929492151137522514627875, 1.90768870369939596815673842215, 3.63865401985788809648207766924, 5.59335591750130596236748420614, 7.66868076472784564979694502445, 8.853859292648564925952270389375, 9.263636361628809552917488189325, 10.64354419191540726245298857130, 11.52966756596120658697734420309, 12.26766474947201923460419782981, 14.02072440547481416313207896316

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