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Properties

Label 2-93-31.2-c1-0-1
Degree $2$
Conductor $93$
Sign $0.998 + 0.0467i$
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  + (0.133 + 0.0971i)2-s + (−0.809 + 0.587i)3-s + (−0.609 − 1.87i)4-s + 3.79·5-s − 0.165·6-s + (0.672 + 2.07i)7-s + (0.202 − 0.624i)8-s + (0.309 − 0.951i)9-s + (0.507 + 0.368i)10-s + (−0.567 − 1.74i)11-s + (1.59 + 1.15i)12-s + (−4.39 + 3.19i)13-s + (−0.111 + 0.342i)14-s + (−3.07 + 2.23i)15-s + (−3.10 + 2.25i)16-s + (1.38 − 4.25i)17-s + ⋯
L(s)  = 1  + (0.0945 + 0.0687i)2-s + (−0.467 + 0.339i)3-s + (−0.304 − 0.938i)4-s + 1.69·5-s − 0.0674·6-s + (0.254 + 0.782i)7-s + (0.0717 − 0.220i)8-s + (0.103 − 0.317i)9-s + (0.160 + 0.116i)10-s + (−0.171 − 0.526i)11-s + (0.460 + 0.334i)12-s + (−1.21 + 0.884i)13-s + (−0.0297 + 0.0914i)14-s + (−0.792 + 0.575i)15-s + (−0.775 + 0.563i)16-s + (0.335 − 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.998 + 0.0467i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1/2),\ 0.998 + 0.0467i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03875 - 0.0242739i\)
\(L(\frac12)\) \(\approx\) \(1.03875 - 0.0242739i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (5.56 + 0.177i)T \)
good2 \( 1 + (-0.133 - 0.0971i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 - 3.79T + 5T^{2} \)
7 \( 1 + (-0.672 - 2.07i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (0.567 + 1.74i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (4.39 - 3.19i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.38 + 4.25i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (4.00 + 2.90i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (2.46 - 7.58i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-3.33 - 2.42i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + 0.574T + 37T^{2} \)
41 \( 1 + (0.507 + 0.368i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-1.01 - 0.734i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-1.43 + 1.04i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.76 + 5.44i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.12 + 4.45i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + 6.32T + 61T^{2} \)
67 \( 1 - 6.33T + 67T^{2} \)
71 \( 1 + (-0.761 + 2.34i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (2.86 + 8.82i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.169 - 0.521i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.206 + 0.150i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-1.79 - 5.51i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-3.28 - 10.1i)T + (-78.4 + 57.0i)T^{2} \)
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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

−14.10809055114666290571914684543, −13.24970954751797245416193547698, −11.81971004313072614011006691405, −10.61457231517928414904364420351, −9.571665894440781325232581296136, −9.139917755866402861057834188330, −6.72694022880285808732153852380, −5.58887775024161797340002965740, −5.01483605655979256670496113126, −2.10489617996726533401048379004, 2.29377483756931696582143714584, 4.50986892997984877617996267013, 5.87542560067510540451190296762, 7.20543742650232665510782395216, 8.400613296585965174483972775808, 9.990186568742539833236810342290, 10.57512523441546088423562785194, 12.50921770371572751284916024838, 12.73066535982059543072216348240, 13.88552563952354390397428166238

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