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Properties

Label 2-93-93.29-c1-0-3
Degree $2$
Conductor $93$
Sign $0.997 - 0.0688i$
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 + (0.0480 + 1.73i)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 + (−2.99 + 0.166i)9-s + (0.0968 + 0.0703i)10-s + (−0.770 − 2.37i)11-s + (−0.892 + 0.317i)12-s + (−1.72 − 2.37i)13-s + (−2.67 − 0.868i)14-s + (−0.171 + 0.00477i)15-s + (2.10 − 1.53i)16-s + (−0.143 + 0.441i)17-s + ⋯
L(s)  = 1  + (0.500 − 0.689i)2-s + (0.0277 + 0.999i)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 + (−0.998 + 0.0554i)9-s + (0.0306 + 0.0222i)10-s + (−0.232 − 0.715i)11-s + (−0.257 + 0.0917i)12-s + (−0.478 − 0.658i)13-s + (−0.714 − 0.232i)14-s + (−0.0444 + 0.00123i)15-s + (0.527 − 0.383i)16-s + (−0.0347 + 0.107i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.997 - 0.0688i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1/2),\ 0.997 - 0.0688i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25279 + 0.0431573i\)
\(L(\frac12)\) \(\approx\) \(1.25279 + 0.0431573i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0480 - 1.73i)T \)
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

−13.85260593679982538557251739110, −13.07221910072078979706870820286, −11.78815839384025531827417392460, −10.79774953504935819065787124789, −10.17343867147631903207276078291, −8.657458304897911833251413164679, −7.33911580611861887410598754950, −5.39498545008185344515097070903, −4.04111336641592022300794460429, −3.01619032287773744984314437895, 2.22275018512443219012179568738, 4.79259730804999905623581349856, 6.11896073043502117392720476769, 6.89941993631247123922542161253, 8.132019879972987429070625374282, 9.523217240986746237913362594180, 10.99038202239959866726879018223, 12.38389946372169434663145642574, 12.90373952361799379323297081817, 14.35274042454083443943612208925

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