Properties

Label 2-93-93.44-c1-0-1
Degree $2$
Conductor $93$
Sign $0.957 - 0.289i$
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  + (0.704 + 1.58i)3-s + (0.618 − 1.90i)4-s + (0.802 + 0.891i)7-s + (−2.00 + 2.22i)9-s + (3.44 − 0.362i)12-s + (−3.13 − 0.329i)13-s + (−3.23 − 2.35i)16-s + (−0.742 − 7.06i)19-s + (−0.845 + 1.89i)21-s + (−2.5 + 4.33i)25-s + (−4.94 − 1.60i)27-s + (2.19 − 0.976i)28-s + (−4.67 + 3.02i)31-s + (2.99 + 5.19i)36-s + (9.23 + 5.33i)37-s + ⋯
L(s)  = 1  + (0.406 + 0.913i)3-s + (0.309 − 0.951i)4-s + (0.303 + 0.337i)7-s + (−0.669 + 0.743i)9-s + (0.994 − 0.104i)12-s + (−0.868 − 0.0913i)13-s + (−0.809 − 0.587i)16-s + (−0.170 − 1.62i)19-s + (−0.184 + 0.414i)21-s + (−0.5 + 0.866i)25-s + (−0.951 − 0.309i)27-s + (0.414 − 0.184i)28-s + (−0.839 + 0.543i)31-s + (0.499 + 0.866i)36-s + (1.51 + 0.877i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.957 - 0.289i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1/2),\ 0.957 - 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12087 + 0.165662i\)
\(L(\frac12)\) \(\approx\) \(1.12087 + 0.165662i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.704 - 1.58i)T \)
31 \( 1 + (4.67 - 3.02i)T \)
good2 \( 1 + (-0.618 + 1.90i)T^{2} \)
5 \( 1 + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.802 - 0.891i)T + (-0.731 + 6.96i)T^{2} \)
11 \( 1 + (10.0 - 4.47i)T^{2} \)
13 \( 1 + (3.13 + 0.329i)T + (12.7 + 2.70i)T^{2} \)
17 \( 1 + (15.5 + 6.91i)T^{2} \)
19 \( 1 + (0.742 + 7.06i)T + (-18.5 + 3.95i)T^{2} \)
23 \( 1 + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (8.96 - 27.5i)T^{2} \)
37 \( 1 + (-9.23 - 5.33i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-27.4 - 30.4i)T^{2} \)
43 \( 1 + (-12.6 + 1.33i)T + (42.0 - 8.94i)T^{2} \)
47 \( 1 + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-5.54 - 52.7i)T^{2} \)
59 \( 1 + (-39.4 + 43.8i)T^{2} \)
61 \( 1 - 15.4iT - 61T^{2} \)
67 \( 1 + (0.824 + 1.42i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.42 + 70.6i)T^{2} \)
73 \( 1 + (-1.78 - 8.40i)T + (-66.6 + 29.6i)T^{2} \)
79 \( 1 + (-1.90 + 8.95i)T + (-72.1 - 32.1i)T^{2} \)
83 \( 1 + (55.5 + 61.6i)T^{2} \)
89 \( 1 + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-6.07 + 18.7i)T + (-78.4 - 57.0i)T^{2} \)
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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

−14.44232717718943196056818401246, −13.29361821835728431264634480135, −11.60408495778892491716092158428, −10.80355605885503702916418724485, −9.714634593925287765181042462338, −8.921310733395721289221186920020, −7.31047869067334859464621886682, −5.63839884395016422868469324249, −4.62180512863275309057174678269, −2.54668429802209106816778612373, 2.30832901970724047000910561847, 3.95735900805063109629136221088, 6.14923387345495093590831470243, 7.52776768865799348607162048739, 7.998371601507813529583253134841, 9.410544654706678571146070353200, 11.05252867016766539524692367610, 12.21368963798446889128065722891, 12.71443383092426886876253865337, 13.96333408809866158237468210493

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