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Properties

Label 2-58-29.24-c1-0-1
Degree $2$
Conductor $58$
Sign $0.645 - 0.763i$
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 + (0.260 + 1.14i)3-s + (−0.222 + 0.974i)4-s + (−1.85 − 2.32i)5-s + (−0.729 + 0.915i)6-s + (−0.115 − 0.507i)7-s + (−0.900 + 0.433i)8-s + (1.46 − 0.707i)9-s + (0.661 − 2.89i)10-s + (0.585 + 0.281i)11-s − 1.17·12-s + (−0.444 − 0.214i)13-s + (0.324 − 0.407i)14-s + (2.16 − 2.72i)15-s + (−0.900 − 0.433i)16-s − 7.42·17-s + ⋯
L(s)  = 1  + (0.440 + 0.552i)2-s + (0.150 + 0.658i)3-s + (−0.111 + 0.487i)4-s + (−0.828 − 1.03i)5-s + (−0.297 + 0.373i)6-s + (−0.0438 − 0.191i)7-s + (−0.318 + 0.153i)8-s + (0.489 − 0.235i)9-s + (0.209 − 0.916i)10-s + (0.176 + 0.0849i)11-s − 0.337·12-s + (−0.123 − 0.0593i)13-s + (0.0868 − 0.108i)14-s + (0.560 − 0.702i)15-s + (−0.225 − 0.108i)16-s − 1.79·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 - 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58\)    =    \(2 \cdot 29\)
Sign: $0.645 - 0.763i$
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: \((2,\ 58,\ (\ :1/2),\ 0.645 - 0.763i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.900264 + 0.418042i\)
\(L(\frac12)\) \(\approx\) \(0.900264 + 0.418042i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.56 - 2.85i)T \)
good3 \( 1 + (-0.260 - 1.14i)T + (-2.70 + 1.30i)T^{2} \)
5 \( 1 + (1.85 + 2.32i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 + (0.115 + 0.507i)T + (-6.30 + 3.03i)T^{2} \)
11 \( 1 + (-0.585 - 0.281i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (0.444 + 0.214i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + 7.42T + 17T^{2} \)
19 \( 1 + (1.47 - 6.46i)T + (-17.1 - 8.24i)T^{2} \)
23 \( 1 + (-4.74 + 5.95i)T + (-5.11 - 22.4i)T^{2} \)
31 \( 1 + (-3.72 - 4.67i)T + (-6.89 + 30.2i)T^{2} \)
37 \( 1 + (-2.23 + 1.07i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 - 7.82T + 41T^{2} \)
43 \( 1 + (-0.404 + 0.507i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (7.92 + 3.81i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (0.717 + 0.900i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + 5.31T + 59T^{2} \)
61 \( 1 + (2.15 + 9.45i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (4.07 - 1.96i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (-12.7 - 6.13i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-5.44 + 6.83i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + (3.71 - 1.79i)T + (49.2 - 61.7i)T^{2} \)
83 \( 1 + (0.952 - 4.17i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (0.743 + 0.932i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + (2.61 - 11.4i)T + (-87.3 - 42.0i)T^{2} \)
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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

−15.44915212816745930905365453393, −14.54522558447414330623295673075, −12.99593245009247590724379676553, −12.32155508583228960540224570807, −10.80888645103103677591097460725, −9.215370751684651283530695492801, −8.263858928935091364030290934909, −6.74954856336868790242221032863, −4.79962161119483935944837319547, −3.96787236268700373413171601004, 2.59676443865007709016019987668, 4.36684241440737050875581389347, 6.56412772847224890692478856985, 7.52419643212602457948554517108, 9.256816601096349996629778543317, 10.96524064624856343494786479097, 11.47058345750294246324821181064, 12.95492102809106755341650088093, 13.63239781869045746813992642049, 15.12345588823903598461428319101

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