Properties

Label 2-58-29.5-c1-0-1
Degree $2$
Conductor $58$
Sign $0.856 + 0.516i$
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 + (−0.892 − 1.11i)5-s + (−0.153 + 0.193i)6-s + (0.904 + 3.96i)7-s + (−0.433 − 0.900i)8-s + (−2.64 + 1.27i)9-s + (−1.39 − 0.318i)10-s + (0.815 − 1.69i)11-s + 0.246i·12-s + (−3.39 − 1.63i)13-s + (3.17 + 2.53i)14-s + (0.276 + 0.220i)15-s + (−0.900 − 0.433i)16-s + 1.78i·17-s + ⋯
L(s)  = 1  + (0.552 − 0.440i)2-s + (−0.139 + 0.0317i)3-s + (0.111 − 0.487i)4-s + (−0.399 − 0.500i)5-s + (−0.0628 + 0.0788i)6-s + (0.341 + 1.49i)7-s + (−0.153 − 0.318i)8-s + (−0.882 + 0.425i)9-s + (−0.441 − 0.100i)10-s + (0.245 − 0.510i)11-s + 0.0712i·12-s + (−0.942 − 0.453i)13-s + (0.849 + 0.677i)14-s + (0.0713 + 0.0569i)15-s + (−0.225 − 0.108i)16-s + 0.433i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58\)    =    \(2 \cdot 29\)
Sign: $0.856 + 0.516i$
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: \((2,\ 58,\ (\ :1/2),\ 0.856 + 0.516i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.974222 - 0.271240i\)
\(L(\frac12)\) \(\approx\) \(0.974222 - 0.271240i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.25 + 5.23i)T \)
good3 \( 1 + (0.240 - 0.0549i)T + (2.70 - 1.30i)T^{2} \)
5 \( 1 + (0.892 + 1.11i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 + (-0.904 - 3.96i)T + (-6.30 + 3.03i)T^{2} \)
11 \( 1 + (-0.815 + 1.69i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (3.39 + 1.63i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 - 1.78iT - 17T^{2} \)
19 \( 1 + (-3.79 - 0.866i)T + (17.1 + 8.24i)T^{2} \)
23 \( 1 + (-2.97 + 3.73i)T + (-5.11 - 22.4i)T^{2} \)
31 \( 1 + (5.35 - 4.27i)T + (6.89 - 30.2i)T^{2} \)
37 \( 1 + (-2.21 - 4.59i)T + (-23.0 + 28.9i)T^{2} \)
41 \( 1 + 10.2iT - 41T^{2} \)
43 \( 1 + (-2.35 - 1.87i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (5.65 - 11.7i)T + (-29.3 - 36.7i)T^{2} \)
53 \( 1 + (-5.32 - 6.68i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + 5.64T + 59T^{2} \)
61 \( 1 + (11.9 - 2.72i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 + (-10.6 + 5.11i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (3.36 + 1.62i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-3.62 - 2.89i)T + (16.2 + 71.1i)T^{2} \)
79 \( 1 + (-4.80 - 9.98i)T + (-49.2 + 61.7i)T^{2} \)
83 \( 1 + (-0.807 + 3.53i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-1.33 + 1.06i)T + (19.8 - 86.7i)T^{2} \)
97 \( 1 + (-12.5 - 2.85i)T + (87.3 + 42.0i)T^{2} \)
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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

−14.96664688738473047174439163594, −14.05595015697099887679366195930, −12.48878237337847249358086710617, −11.96053912065093885384769798561, −10.85588483199967557541452303918, −9.173441127654296674348032785870, −8.111537802088192658706843000566, −5.89775981222853185465821573931, −4.91520109101563756724897059359, −2.73890854034941633941409857341, 3.50494117713141257807678564330, 5.02504750062218568043759937443, 6.92488060557722987718663293692, 7.55785216898248995905648137891, 9.444531856165966312099997785901, 11.03406547547535347751772790025, 11.82814273938595433754477327037, 13.30874616963177766845818784411, 14.38534331772348778704372950040, 14.92447152340280798076639636446

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