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Properties

Label 2-58-29.20-c1-0-1
Degree $2$
Conductor $58$
Sign $0.497 + 0.867i$
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 + (−1.52 − 1.90i)3-s + (0.623 − 0.781i)4-s + (2.60 − 1.25i)5-s + (2.19 + 1.05i)6-s + (−1.89 − 2.37i)7-s + (−0.222 + 0.974i)8-s + (−0.658 + 2.88i)9-s + (−1.79 + 2.25i)10-s + (1.22 + 5.34i)11-s − 2.44·12-s + (0.0239 + 0.104i)13-s + (2.74 + 1.32i)14-s + (−6.34 − 3.05i)15-s + (−0.222 − 0.974i)16-s + 0.816·17-s + ⋯
L(s)  = 1  + (−0.637 + 0.306i)2-s + (−0.878 − 1.10i)3-s + (0.311 − 0.390i)4-s + (1.16 − 0.559i)5-s + (0.897 + 0.432i)6-s + (−0.717 − 0.899i)7-s + (−0.0786 + 0.344i)8-s + (−0.219 + 0.961i)9-s + (−0.568 + 0.713i)10-s + (0.367 + 1.61i)11-s − 0.704·12-s + (0.00663 + 0.0290i)13-s + (0.732 + 0.352i)14-s + (−1.63 − 0.789i)15-s + (−0.0556 − 0.243i)16-s + 0.197·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58\)    =    \(2 \cdot 29\)
Sign: $0.497 + 0.867i$
Analytic conductor: \(0.463132\)
Root analytic conductor: \(0.680538\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{58} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 58,\ (\ :1/2),\ 0.497 + 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.505998 - 0.292961i\)
\(L(\frac12)\) \(\approx\) \(0.505998 - 0.292961i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.37 + 0.304i)T \)
good3 \( 1 + (1.52 + 1.90i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (-2.60 + 1.25i)T + (3.11 - 3.90i)T^{2} \)
7 \( 1 + (1.89 + 2.37i)T + (-1.55 + 6.82i)T^{2} \)
11 \( 1 + (-1.22 - 5.34i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (-0.0239 - 0.104i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 - 0.816T + 17T^{2} \)
19 \( 1 + (-1.27 + 1.59i)T + (-4.22 - 18.5i)T^{2} \)
23 \( 1 + (-8.25 - 3.97i)T + (14.3 + 17.9i)T^{2} \)
31 \( 1 + (-3.10 + 1.49i)T + (19.3 - 24.2i)T^{2} \)
37 \( 1 + (1.31 - 5.75i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + 2.43T + 41T^{2} \)
43 \( 1 + (4.94 + 2.37i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (1.31 + 5.76i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (3.55 - 1.71i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 - 1.13T + 59T^{2} \)
61 \( 1 + (-5.09 - 6.38i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + (0.212 - 0.932i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (0.531 + 2.32i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (8.01 + 3.85i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 + (-0.934 + 4.09i)T + (-71.1 - 34.2i)T^{2} \)
83 \( 1 + (9.64 - 12.0i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (4.99 - 2.40i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + (-7.43 + 9.32i)T + (-21.5 - 94.5i)T^{2} \)
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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

−15.13088816902901066764033821741, −13.47313295606733456371548808576, −12.94330913951854852471480860346, −11.68122478336663530211977701313, −10.14183016051719082352994796072, −9.324038968592998818674907696742, −7.28708893006793575825702581359, −6.66206929019848200283755473434, −5.26531205183255138421576235460, −1.46253132642499295750824365909, 3.11927520482895634263698173579, 5.54087034181358025019017811898, 6.39266598552677948046117087052, 8.861291556988982830764991044098, 9.714271284552490859378242950739, 10.68450130722518723639181048828, 11.49272491730028602910923243900, 12.98611579061472976487609281449, 14.43362710956709640046237494933, 15.76864140591131721085978036823

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