Properties

Label 2-58-29.22-c1-0-0
Degree $2$
Conductor $58$
Sign $0.987 - 0.155i$
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  + (0.974 − 0.222i)2-s + (−0.626 + 1.30i)3-s + (0.900 − 0.433i)4-s + (−0.136 − 0.598i)5-s + (−0.321 + 1.40i)6-s + (−2.59 − 1.24i)7-s + (0.781 − 0.623i)8-s + (0.568 + 0.712i)9-s + (−0.266 − 0.553i)10-s + (−3.39 − 2.70i)11-s + 1.44i·12-s + (−0.298 + 0.373i)13-s + (−2.80 − 0.640i)14-s + (0.864 + 0.197i)15-s + (0.623 − 0.781i)16-s + 0.259i·17-s + ⋯
L(s)  = 1  + (0.689 − 0.157i)2-s + (−0.361 + 0.751i)3-s + (0.450 − 0.216i)4-s + (−0.0610 − 0.267i)5-s + (−0.131 + 0.575i)6-s + (−0.979 − 0.471i)7-s + (0.276 − 0.220i)8-s + (0.189 + 0.237i)9-s + (−0.0842 − 0.174i)10-s + (−1.02 − 0.816i)11-s + 0.417i·12-s + (−0.0827 + 0.103i)13-s + (−0.749 − 0.171i)14-s + (0.223 + 0.0509i)15-s + (0.155 − 0.195i)16-s + 0.0629i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58\)    =    \(2 \cdot 29\)
Sign: $0.987 - 0.155i$
Analytic conductor: \(0.463132\)
Root analytic conductor: \(0.680538\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{58} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 58,\ (\ :1/2),\ 0.987 - 0.155i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03668 + 0.0810499i\)
\(L(\frac12)\) \(\approx\) \(1.03668 + 0.0810499i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.974 + 0.222i)T \)
29 \( 1 + (-1.07 + 5.27i)T \)
good3 \( 1 + (0.626 - 1.30i)T + (-1.87 - 2.34i)T^{2} \)
5 \( 1 + (0.136 + 0.598i)T + (-4.50 + 2.16i)T^{2} \)
7 \( 1 + (2.59 + 1.24i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (3.39 + 2.70i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (0.298 - 0.373i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 - 0.259iT - 17T^{2} \)
19 \( 1 + (-3.65 - 7.58i)T + (-11.8 + 14.8i)T^{2} \)
23 \( 1 + (-0.0317 + 0.139i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (-6.46 + 1.47i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + (7.50 - 5.98i)T + (8.23 - 36.0i)T^{2} \)
41 \( 1 - 4.28iT - 41T^{2} \)
43 \( 1 + (-3.17 - 0.725i)T + (38.7 + 18.6i)T^{2} \)
47 \( 1 + (3.97 + 3.16i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (2.06 + 9.03i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + 10.2T + 59T^{2} \)
61 \( 1 + (4.31 - 8.95i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 + (1.16 + 1.45i)T + (-14.9 + 65.3i)T^{2} \)
71 \( 1 + (-5.97 + 7.49i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-2.90 - 0.663i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + (-10.5 + 8.45i)T + (17.5 - 77.0i)T^{2} \)
83 \( 1 + (0.950 - 0.457i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-2.51 + 0.573i)T + (80.1 - 38.6i)T^{2} \)
97 \( 1 + (7.22 + 14.9i)T + (-60.4 + 75.8i)T^{2} \)
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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

−15.44068743548308717039502450521, −13.90858565666793438404746944825, −13.10144011652486533996868299351, −11.89725038594664561555602162736, −10.51125133700451527989885648245, −9.914657507020135394067687525949, −7.939680651948477430103996188402, −6.18142973007770478003668890862, −4.88259194769045220905788789861, −3.40073848243964366762658380321, 2.89830803448797373133131756996, 5.11455510042693819454772250834, 6.60222041131859291984605478493, 7.37020028640663026277454248835, 9.347214670971176741664107726377, 10.82973288710690880389291537270, 12.29302597356822134828674654694, 12.75829180558022217901394075965, 13.81708802171082540181361986514, 15.40685340002678857054441572925

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