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Properties

Label 2-93-93.77-c1-0-6
Degree $2$
Conductor $93$
Sign $0.296 + 0.955i$
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  + (−0.708 − 0.975i)2-s + (1.63 − 0.580i)3-s + (0.169 − 0.520i)4-s + 0.0993i·5-s + (−1.72 − 1.17i)6-s + (−0.720 + 2.21i)7-s + (−2.91 + 0.948i)8-s + (2.32 − 1.89i)9-s + (0.0968 − 0.0703i)10-s + (0.770 − 2.37i)11-s + (−0.0262 − 0.947i)12-s + (−1.72 + 2.37i)13-s + (2.67 − 0.868i)14-s + (0.0576 + 0.162i)15-s + (2.10 + 1.53i)16-s + (0.143 + 0.441i)17-s + ⋯
L(s)  = 1  + (−0.500 − 0.689i)2-s + (0.942 − 0.335i)3-s + (0.0845 − 0.260i)4-s + 0.0444i·5-s + (−0.703 − 0.481i)6-s + (−0.272 + 0.837i)7-s + (−1.03 + 0.335i)8-s + (0.775 − 0.631i)9-s + (0.0306 − 0.0222i)10-s + (0.232 − 0.715i)11-s + (−0.00759 − 0.273i)12-s + (−0.478 + 0.658i)13-s + (0.714 − 0.232i)14-s + (0.0148 + 0.0418i)15-s + (0.527 + 0.383i)16-s + (0.0347 + 0.107i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.296 + 0.955i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1/2),\ 0.296 + 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.806288 - 0.594066i\)
\(L(\frac12)\) \(\approx\) \(0.806288 - 0.594066i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.63 + 0.580i)T \)
31 \( 1 + (5.55 + 0.356i)T \)
good2 \( 1 + (0.708 + 0.975i)T + (-0.618 + 1.90i)T^{2} \)
5 \( 1 - 0.0993iT - 5T^{2} \)
7 \( 1 + (0.720 - 2.21i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (-0.770 + 2.37i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.72 - 2.37i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.143 - 0.441i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.94 - 1.41i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-2.36 - 7.27i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (2.51 - 1.82i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 + 4.25iT - 37T^{2} \)
41 \( 1 + (6.25 + 8.61i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (1.56 + 2.14i)T + (-13.2 + 40.8i)T^{2} \)
47 \( 1 + (-6.38 + 8.78i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.781 - 2.40i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (8.22 - 11.3i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + 13.7iT - 61T^{2} \)
67 \( 1 + 9.63T + 67T^{2} \)
71 \( 1 + (4.50 - 1.46i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-3.40 - 1.10i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.69 - 1.52i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.452 - 0.328i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-3.79 + 11.6i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-2.73 + 8.41i)T + (-78.4 - 57.0i)T^{2} \)
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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

−13.88435887929499620215918461398, −12.59874598087798046065834545640, −11.71871504859984152501572205163, −10.48732430142342290189112345784, −9.163573192080042147472629155893, −8.859386624710725375811769058253, −7.14238548619193225008911355814, −5.71243077164622627265891174796, −3.34993574485263597712023326253, −1.93505429270726303284286188322, 2.97861512695196830862576116362, 4.51342858780308219741092859763, 6.74289172545805493746846128110, 7.56868533829153399759985205653, 8.621202880057609273378849394005, 9.631641780312370771529918271206, 10.66313321562241240055800967013, 12.46123105553155363790679434121, 13.23046870915769534349780010832, 14.68059205277021091507142171563

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