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Properties

Label 2-93-93.92-c3-0-26
Degree $2$
Conductor $93$
Sign $-0.900 - 0.433i$
Analytic cond. $5.48717$
Root an. cond. $2.34247$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.24i·2-s + (5.08 − 1.08i)3-s − 19.5·4-s − 7.60i·5-s + (−5.67 − 26.6i)6-s − 11.1·7-s + 60.3i·8-s + (24.6 − 10.9i)9-s − 39.8·10-s − 35.9·11-s + (−99.1 + 21.0i)12-s − 45.7i·13-s + 58.4i·14-s + (−8.22 − 38.6i)15-s + 160.·16-s + 126.·17-s + ⋯
L(s)  = 1  − 1.85i·2-s + (0.978 − 0.208i)3-s − 2.43·4-s − 0.680i·5-s + (−0.385 − 1.81i)6-s − 0.601·7-s + 2.66i·8-s + (0.913 − 0.407i)9-s − 1.26·10-s − 0.984·11-s + (−2.38 + 0.507i)12-s − 0.975i·13-s + 1.11i·14-s + (−0.141 − 0.665i)15-s + 2.50·16-s + 1.79·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.433i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $-0.900 - 0.433i$
Analytic conductor: \(5.48717\)
Root analytic conductor: \(2.34247\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :3/2),\ -0.900 - 0.433i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.344942 + 1.51090i\)
\(L(\frac12)\) \(\approx\) \(0.344942 + 1.51090i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.08 + 1.08i)T \)
31 \( 1 + (136. + 105. i)T \)
good2 \( 1 + 5.24iT - 8T^{2} \)
5 \( 1 + 7.60iT - 125T^{2} \)
7 \( 1 + 11.1T + 343T^{2} \)
11 \( 1 + 35.9T + 1.33e3T^{2} \)
13 \( 1 + 45.7iT - 2.19e3T^{2} \)
17 \( 1 - 126.T + 4.91e3T^{2} \)
19 \( 1 + 41.2T + 6.85e3T^{2} \)
23 \( 1 - 13.0T + 1.21e4T^{2} \)
29 \( 1 - 161.T + 2.43e4T^{2} \)
37 \( 1 - 135. iT - 5.06e4T^{2} \)
41 \( 1 + 393. iT - 6.89e4T^{2} \)
43 \( 1 + 421. iT - 7.95e4T^{2} \)
47 \( 1 - 538. iT - 1.03e5T^{2} \)
53 \( 1 - 174.T + 1.48e5T^{2} \)
59 \( 1 + 506. iT - 2.05e5T^{2} \)
61 \( 1 - 695. iT - 2.26e5T^{2} \)
67 \( 1 - 273.T + 3.00e5T^{2} \)
71 \( 1 - 188. iT - 3.57e5T^{2} \)
73 \( 1 - 882. iT - 3.89e5T^{2} \)
79 \( 1 - 400. iT - 4.93e5T^{2} \)
83 \( 1 - 805.T + 5.71e5T^{2} \)
89 \( 1 - 1.24e3T + 7.04e5T^{2} \)
97 \( 1 + 157.T + 9.12e5T^{2} \)
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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

−12.71143047107423763171659114504, −12.29377320501157058935438681748, −10.53514590265079922526944686556, −9.901197090535873761281581021951, −8.819714142939869282661862894015, −7.86824274130383537931996250096, −5.21144126012142401752271940704, −3.63496805545868275952811415390, −2.61561006128865883567669580652, −0.878479822952819950094748250564, 3.30882563354571123988728622383, 4.88646577261010147726835802907, 6.41161519335628214368243359961, 7.38458391375037858532374693536, 8.272018259598993014742787386494, 9.419979423765736040178163064873, 10.32215416315257773496822698071, 12.68009926381263091475970587990, 13.64551506990922317913514564104, 14.52257569955149561484472359184

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