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Properties

Label 2-58-29.24-c1-0-2
Degree $2$
Conductor $58$
Sign $0.946 + 0.323i$
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  + (0.623 + 0.781i)2-s + (−0.760 − 3.33i)3-s + (−0.222 + 0.974i)4-s + (1.00 + 1.26i)5-s + (2.13 − 2.67i)6-s + (0.338 + 1.48i)7-s + (−0.900 + 0.433i)8-s + (−7.81 + 3.76i)9-s + (−0.359 + 1.57i)10-s + (−1.70 − 0.822i)11-s + 3.41·12-s + (2.87 + 1.38i)13-s + (−0.948 + 1.18i)14-s + (3.44 − 4.31i)15-s + (−0.900 − 0.433i)16-s − 1.69·17-s + ⋯
L(s)  = 1  + (0.440 + 0.552i)2-s + (−0.439 − 1.92i)3-s + (−0.111 + 0.487i)4-s + (0.450 + 0.564i)5-s + (0.869 − 1.09i)6-s + (0.127 + 0.560i)7-s + (−0.318 + 0.153i)8-s + (−2.60 + 1.25i)9-s + (−0.113 + 0.498i)10-s + (−0.515 − 0.248i)11-s + 0.986·12-s + (0.796 + 0.383i)13-s + (−0.253 + 0.317i)14-s + (0.888 − 1.11i)15-s + (−0.225 − 0.108i)16-s − 0.412·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58\)    =    \(2 \cdot 29\)
Sign: $0.946 + 0.323i$
Analytic conductor: \(0.463132\)
Root analytic conductor: \(0.680538\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{58} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 58,\ (\ :1/2),\ 0.946 + 0.323i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.936571 - 0.155466i\)
\(L(\frac12)\) \(\approx\) \(0.936571 - 0.155466i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.623 - 0.781i)T \)
29 \( 1 + (4.31 + 3.21i)T \)
good3 \( 1 + (0.760 + 3.33i)T + (-2.70 + 1.30i)T^{2} \)
5 \( 1 + (-1.00 - 1.26i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 + (-0.338 - 1.48i)T + (-6.30 + 3.03i)T^{2} \)
11 \( 1 + (1.70 + 0.822i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (-2.87 - 1.38i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + 1.69T + 17T^{2} \)
19 \( 1 + (-0.818 + 3.58i)T + (-17.1 - 8.24i)T^{2} \)
23 \( 1 + (1.67 - 2.10i)T + (-5.11 - 22.4i)T^{2} \)
31 \( 1 + (1.11 + 1.39i)T + (-6.89 + 30.2i)T^{2} \)
37 \( 1 + (-6.37 + 3.06i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + 5.03T + 41T^{2} \)
43 \( 1 + (1.18 - 1.48i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (2.31 + 1.11i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-5.14 - 6.44i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 - 2.95T + 59T^{2} \)
61 \( 1 + (-1.72 - 7.55i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (2.23 - 1.07i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (5.46 + 2.63i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-2.33 + 2.93i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + (8.30 - 4.00i)T + (49.2 - 61.7i)T^{2} \)
83 \( 1 + (-1.22 + 5.38i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-9.95 - 12.4i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + (3.31 - 14.5i)T + (-87.3 - 42.0i)T^{2} \)
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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

−14.85680511843847365626233446937, −13.61551359110241979374469391427, −13.27369502933560177032990902367, −11.96375786599501079344875192884, −11.09563800657855884213056084257, −8.702579319049014101599295326404, −7.52648505793808743685522431452, −6.43749940039944252757714314594, −5.62337168461694740137725333271, −2.42782369973587164779305623392, 3.59636949026535389544487755160, 4.81328645179728076489521002190, 5.82175864823020080384906413696, 8.674825510857470128774381844907, 9.846509652250526694674302949005, 10.57871708585292038434328241430, 11.51593759982589709221455998806, 13.01493683523189819601538549079, 14.28550605011879413666191948013, 15.28435142079124380073279522688

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