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Properties

Label 2-58-29.16-c1-0-2
Degree $2$
Conductor $58$
Sign $0.991 - 0.132i$
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  + (0.900 + 0.433i)2-s + (0.5 − 0.626i)3-s + (0.623 + 0.781i)4-s + (−2.92 − 1.40i)5-s + (0.722 − 0.347i)6-s + (−1.62 + 2.03i)7-s + (0.222 + 0.974i)8-s + (0.524 + 2.29i)9-s + (−2.02 − 2.53i)10-s + (1.20 − 5.27i)11-s + 0.801·12-s + (−0.425 + 1.86i)13-s + (−2.34 + 1.12i)14-s + (−2.34 + 1.12i)15-s + (−0.222 + 0.974i)16-s + 2.85·17-s + ⋯
L(s)  = 1  + (0.637 + 0.306i)2-s + (0.288 − 0.361i)3-s + (0.311 + 0.390i)4-s + (−1.30 − 0.630i)5-s + (0.294 − 0.142i)6-s + (−0.613 + 0.769i)7-s + (0.0786 + 0.344i)8-s + (0.174 + 0.765i)9-s + (−0.640 − 0.802i)10-s + (0.362 − 1.58i)11-s + 0.231·12-s + (−0.117 + 0.516i)13-s + (−0.626 + 0.301i)14-s + (−0.605 + 0.291i)15-s + (−0.0556 + 0.243i)16-s + 0.691·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58\)    =    \(2 \cdot 29\)
Sign: $0.991 - 0.132i$
Analytic conductor: \(0.463132\)
Root analytic conductor: \(0.680538\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{58} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 58,\ (\ :1/2),\ 0.991 - 0.132i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06934 + 0.0713054i\)
\(L(\frac12)\) \(\approx\) \(1.06934 + 0.0713054i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.900 - 0.433i)T \)
29 \( 1 + (-3.51 - 4.08i)T \)
good3 \( 1 + (-0.5 + 0.626i)T + (-0.667 - 2.92i)T^{2} \)
5 \( 1 + (2.92 + 1.40i)T + (3.11 + 3.90i)T^{2} \)
7 \( 1 + (1.62 - 2.03i)T + (-1.55 - 6.82i)T^{2} \)
11 \( 1 + (-1.20 + 5.27i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (0.425 - 1.86i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 - 2.85T + 17T^{2} \)
19 \( 1 + (3.20 + 4.01i)T + (-4.22 + 18.5i)T^{2} \)
23 \( 1 + (-1.82 + 0.879i)T + (14.3 - 17.9i)T^{2} \)
31 \( 1 + (1.64 + 0.793i)T + (19.3 + 24.2i)T^{2} \)
37 \( 1 + (-1.50 - 6.59i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + 9.78T + 41T^{2} \)
43 \( 1 + (-2.33 + 1.12i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-0.5 + 2.19i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (11.1 + 5.38i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
61 \( 1 + (-5.56 + 6.97i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 + (-2.44 - 10.7i)T + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (-1.40 + 6.16i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-8.39 + 4.04i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 + (1.69 + 7.41i)T + (-71.1 + 34.2i)T^{2} \)
83 \( 1 + (-4.76 - 5.97i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-7.30 - 3.51i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (-1.41 - 1.77i)T + (-21.5 + 94.5i)T^{2} \)
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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

−15.32406629069645180777816471227, −14.02676507765636239060397527447, −12.98699067495491571950032552849, −12.07251941735634036983992088470, −11.08491854985771172167240069480, −8.834449628372851450842534311419, −8.066082294227148422261423416046, −6.58549716440491780741281232975, −4.91637049011303650429150678343, −3.25524393388551810454495766471, 3.42700433531875783992323969808, 4.28824881609124735563462457205, 6.64979648222450654618329047919, 7.67839627775946668218558221931, 9.703875052091361020824766598349, 10.57782706993428287783150536419, 12.03191909612049620753784904724, 12.66848861020698580614275911208, 14.36179882396978042630245016545, 15.07597765246608033266585741804

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