Properties

Label 2-58-29.5-c1-0-0
Degree $2$
Conductor $58$
Sign $0.785 - 0.619i$
Analytic cond. $0.463132$
Root an. cond. $0.680538$
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  + (−0.781 + 0.623i)2-s + (0.240 − 0.0549i)3-s + (0.222 − 0.974i)4-s + (2.13 + 2.68i)5-s + (−0.153 + 0.193i)6-s + (−0.349 − 1.53i)7-s + (0.433 + 0.900i)8-s + (−2.64 + 1.27i)9-s + (−3.34 − 0.763i)10-s + (1.89 − 3.94i)11-s − 0.246i·12-s + (−4.09 − 1.97i)13-s + (1.22 + 0.979i)14-s + (0.662 + 0.528i)15-s + (−0.900 − 0.433i)16-s − 2.21i·17-s + ⋯
L(s)  = 1  + (−0.552 + 0.440i)2-s + (0.139 − 0.0317i)3-s + (0.111 − 0.487i)4-s + (0.956 + 1.19i)5-s + (−0.0628 + 0.0788i)6-s + (−0.132 − 0.578i)7-s + (0.153 + 0.318i)8-s + (−0.882 + 0.425i)9-s + (−1.05 − 0.241i)10-s + (0.572 − 1.18i)11-s − 0.0712i·12-s + (−1.13 − 0.546i)13-s + (0.328 + 0.261i)14-s + (0.171 + 0.136i)15-s + (−0.225 − 0.108i)16-s − 0.537i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 - 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.785 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58\)    =    \(2 \cdot 29\)
Sign: $0.785 - 0.619i$
Analytic conductor: \(0.463132\)
Root analytic conductor: \(0.680538\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{58} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 58,\ (\ :1/2),\ 0.785 - 0.619i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.711232 + 0.246647i\)
\(L(\frac12)\) \(\approx\) \(0.711232 + 0.246647i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.781 - 0.623i)T \)
29 \( 1 + (-4.45 - 3.01i)T \)
good3 \( 1 + (-0.240 + 0.0549i)T + (2.70 - 1.30i)T^{2} \)
5 \( 1 + (-2.13 - 2.68i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 + (0.349 + 1.53i)T + (-6.30 + 3.03i)T^{2} \)
11 \( 1 + (-1.89 + 3.94i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (4.09 + 1.97i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + 2.21iT - 17T^{2} \)
19 \( 1 + (0.412 + 0.0941i)T + (17.1 + 8.24i)T^{2} \)
23 \( 1 + (3.83 - 4.81i)T + (-5.11 - 22.4i)T^{2} \)
31 \( 1 + (-5.83 + 4.65i)T + (6.89 - 30.2i)T^{2} \)
37 \( 1 + (-0.689 - 1.43i)T + (-23.0 + 28.9i)T^{2} \)
41 \( 1 - 4.56iT - 41T^{2} \)
43 \( 1 + (7.24 + 5.77i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (1.87 - 3.88i)T + (-29.3 - 36.7i)T^{2} \)
53 \( 1 + (-4.07 - 5.11i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + 14.5T + 59T^{2} \)
61 \( 1 + (-6.00 + 1.37i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 + (7.42 - 3.57i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (-11.4 - 5.49i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-6.75 - 5.38i)T + (16.2 + 71.1i)T^{2} \)
79 \( 1 + (-2.79 - 5.80i)T + (-49.2 + 61.7i)T^{2} \)
83 \( 1 + (0.137 - 0.600i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (3.96 - 3.15i)T + (19.8 - 86.7i)T^{2} \)
97 \( 1 + (7.93 + 1.81i)T + (87.3 + 42.0i)T^{2} \)
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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

−15.20624410897292956066868743410, −14.02359628945667233893742341524, −13.80063732373782304499069458231, −11.55733042309159588584432980261, −10.48953241826900381184327615402, −9.591155828883607619565946530803, −8.062458789284293234215829438429, −6.75628444222962265833722106003, −5.64156195955461463470596668472, −2.83909760014318747492747329762, 2.16443868136155890458723981587, 4.72472166053780635200807459017, 6.38581334099364410204812409356, 8.388013440253059930766473954520, 9.286451139362418462409872537795, 10.04272511268796809338269107884, 12.11240505998814438543875816120, 12.34716191917846009649584595822, 13.89634493200839564118510596355, 15.04142267167938656468780930180

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