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Properties

Label 2-58-29.12-c2-0-3
Degree $2$
Conductor $58$
Sign $-0.154 + 0.988i$
Analytic cond. $1.58038$
Root an. cond. $1.25713$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + (−0.147 − 0.147i)3-s + 2i·4-s − 8.54i·5-s + 0.295i·6-s − 0.295·7-s + (2 − 2i)8-s − 8.95i·9-s + (−8.54 + 8.54i)10-s + (−0.442 − 0.442i)11-s + (0.295 − 0.295i)12-s + 6.54i·13-s + (0.295 + 0.295i)14-s + (−1.26 + 1.26i)15-s − 4·16-s + (8.95 + 8.95i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.0492 − 0.0492i)3-s + 0.5i·4-s − 1.70i·5-s + 0.0492i·6-s − 0.0421·7-s + (0.250 − 0.250i)8-s − 0.995i·9-s + (−0.854 + 0.854i)10-s + (−0.0402 − 0.0402i)11-s + (0.0246 − 0.0246i)12-s + 0.503i·13-s + (0.0210 + 0.0210i)14-s + (−0.0841 + 0.0841i)15-s − 0.250·16-s + (0.526 + 0.526i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 + 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.154 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58\)    =    \(2 \cdot 29\)
Sign: $-0.154 + 0.988i$
Analytic conductor: \(1.58038\)
Root analytic conductor: \(1.25713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{58} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 58,\ (\ :1),\ -0.154 + 0.988i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.576677 - 0.673526i\)
\(L(\frac12)\) \(\approx\) \(0.576677 - 0.673526i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
29 \( 1 + (26.2 - 12.3i)T \)
good3 \( 1 + (0.147 + 0.147i)T + 9iT^{2} \)
5 \( 1 + 8.54iT - 25T^{2} \)
7 \( 1 + 0.295T + 49T^{2} \)
11 \( 1 + (0.442 + 0.442i)T + 121iT^{2} \)
13 \( 1 - 6.54iT - 169T^{2} \)
17 \( 1 + (-8.95 - 8.95i)T + 289iT^{2} \)
19 \( 1 + (-19.9 - 19.9i)T + 361iT^{2} \)
23 \( 1 - 30.1T + 529T^{2} \)
31 \( 1 + (28.9 + 28.9i)T + 961iT^{2} \)
37 \( 1 + (-25.8 + 25.8i)T - 1.36e3iT^{2} \)
41 \( 1 + (-31.7 + 31.7i)T - 1.68e3iT^{2} \)
43 \( 1 + (-50.6 - 50.6i)T + 1.84e3iT^{2} \)
47 \( 1 + (8.98 - 8.98i)T - 2.20e3iT^{2} \)
53 \( 1 - 36.0T + 2.80e3T^{2} \)
59 \( 1 - 17.1T + 3.48e3T^{2} \)
61 \( 1 + (0.499 + 0.499i)T + 3.72e3iT^{2} \)
67 \( 1 - 113. iT - 4.48e3T^{2} \)
71 \( 1 + 80.2iT - 5.04e3T^{2} \)
73 \( 1 + (-7.45 + 7.45i)T - 5.32e3iT^{2} \)
79 \( 1 + (-40.9 - 40.9i)T + 6.24e3iT^{2} \)
83 \( 1 + 104T + 6.88e3T^{2} \)
89 \( 1 + (-43.6 - 43.6i)T + 7.92e3iT^{2} \)
97 \( 1 + (-55.4 + 55.4i)T - 9.40e3iT^{2} \)
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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

−14.61858136936827141306117208714, −12.99201227299513480102769747274, −12.43158289646496061995226188066, −11.36255481706236493767893910875, −9.572621397881206596706801984954, −9.013222949537231786852163600207, −7.66857876481721200958718361485, −5.63180427588050855257209659047, −3.92347475206225640909444950916, −1.15391140507442434004647527706, 2.87526278439503989591921713497, 5.37913960605198300717458070994, 6.96408443919003709028938568203, 7.69101789154214724965280494916, 9.496653190097419979757123431528, 10.66240708688743942675576260855, 11.31886129277607021175986866591, 13.35841387394291212809741046539, 14.36382697296555482385314058420, 15.23582376755031240505541751217

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