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Properties

Label 2-58-29.20-c1-0-0
Degree $2$
Conductor $58$
Sign $0.614 - 0.788i$
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 + (1.02 + 1.28i)3-s + (0.623 − 0.781i)4-s + (−1.07 + 0.517i)5-s + (−1.47 − 0.711i)6-s + (1.27 + 1.59i)7-s + (−0.222 + 0.974i)8-s + (0.0699 − 0.306i)9-s + (0.744 − 0.933i)10-s + (−0.819 − 3.59i)11-s + 1.63·12-s + (−0.479 − 2.10i)13-s + (−1.84 − 0.886i)14-s + (−1.76 − 0.848i)15-s + (−0.222 − 0.974i)16-s − 6.53·17-s + ⋯
L(s)  = 1  + (−0.637 + 0.306i)2-s + (0.589 + 0.739i)3-s + (0.311 − 0.390i)4-s + (−0.481 + 0.231i)5-s + (−0.602 − 0.290i)6-s + (0.481 + 0.603i)7-s + (−0.0786 + 0.344i)8-s + (0.0233 − 0.102i)9-s + (0.235 − 0.295i)10-s + (−0.247 − 1.08i)11-s + 0.473·12-s + (−0.133 − 0.583i)13-s + (−0.492 − 0.236i)14-s + (−0.455 − 0.219i)15-s + (−0.0556 − 0.243i)16-s − 1.58·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 - 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.614 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58\)    =    \(2 \cdot 29\)
Sign: $0.614 - 0.788i$
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: \((2,\ 58,\ (\ :1/2),\ 0.614 - 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.672759 + 0.328599i\)
\(L(\frac12)\) \(\approx\) \(0.672759 + 0.328599i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.75 - 5.09i)T \)
good3 \( 1 + (-1.02 - 1.28i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (1.07 - 0.517i)T + (3.11 - 3.90i)T^{2} \)
7 \( 1 + (-1.27 - 1.59i)T + (-1.55 + 6.82i)T^{2} \)
11 \( 1 + (0.819 + 3.59i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (0.479 + 2.10i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + 6.53T + 17T^{2} \)
19 \( 1 + (-3.31 + 4.15i)T + (-4.22 - 18.5i)T^{2} \)
23 \( 1 + (-5.30 - 2.55i)T + (14.3 + 17.9i)T^{2} \)
31 \( 1 + (8.10 - 3.90i)T + (19.3 - 24.2i)T^{2} \)
37 \( 1 + (0.406 - 1.78i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + 8.32T + 41T^{2} \)
43 \( 1 + (-3.31 - 1.59i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-0.220 - 0.967i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-5.10 + 2.45i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 - 2.94T + 59T^{2} \)
61 \( 1 + (1.12 + 1.41i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + (1.34 - 5.89i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (0.836 + 3.66i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (11.4 + 5.52i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 + (3.14 - 13.7i)T + (-71.1 - 34.2i)T^{2} \)
83 \( 1 + (-1.27 + 1.60i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-14.8 + 7.16i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + (2.86 - 3.59i)T + (-21.5 - 94.5i)T^{2} \)
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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

−15.42405112315332765079705485417, −14.74709454999103112137187076041, −13.34236243955626325862185173298, −11.54532389285365864472748130329, −10.71084515642885466165705077048, −9.147299650041159251404268489018, −8.576169814825708842112054332374, −7.06390817293857783222318047311, −5.20735773581590030516994544317, −3.17937074556836729539283677817, 2.04754554209974826976360601438, 4.39231051012775475224377202971, 7.05503994162231421680785614924, 7.79567785121068913314616494151, 8.963964115106636420596499608253, 10.40627171126045814272768096541, 11.63000226753219377083002232428, 12.76695450116640065944895007194, 13.76181712033674516475855588533, 15.01687231861016030508341883882

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