Properties

Label 2-58-29.13-c1-0-1
Degree $2$
Conductor $58$
Sign $-0.00689 + 0.999i$
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  + (0.433 − 0.900i)2-s + (−2.19 − 1.74i)3-s + (−0.623 − 0.781i)4-s + (1.63 + 0.788i)5-s + (−2.52 + 1.21i)6-s + (0.882 − 1.10i)7-s + (−0.974 + 0.222i)8-s + (1.07 + 4.72i)9-s + (1.42 − 1.13i)10-s + (3.44 + 0.787i)11-s + 2.80i·12-s + (−1.23 + 5.42i)13-s + (−0.614 − 1.27i)14-s + (−2.20 − 4.58i)15-s + (−0.222 + 0.974i)16-s − 7.46i·17-s + ⋯
L(s)  = 1  + (0.306 − 0.637i)2-s + (−1.26 − 1.00i)3-s + (−0.311 − 0.390i)4-s + (0.732 + 0.352i)5-s + (−1.03 + 0.496i)6-s + (0.333 − 0.418i)7-s + (−0.344 + 0.0786i)8-s + (0.359 + 1.57i)9-s + (0.449 − 0.358i)10-s + (1.03 + 0.237i)11-s + 0.808i·12-s + (−0.343 + 1.50i)13-s + (−0.164 − 0.340i)14-s + (−0.570 − 1.18i)15-s + (−0.0556 + 0.243i)16-s − 1.81i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00689 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00689 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58\)    =    \(2 \cdot 29\)
Sign: $-0.00689 + 0.999i$
Analytic conductor: \(0.463132\)
Root analytic conductor: \(0.680538\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{58} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 58,\ (\ :1/2),\ -0.00689 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.551383 - 0.555199i\)
\(L(\frac12)\) \(\approx\) \(0.551383 - 0.555199i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.433 + 0.900i)T \)
29 \( 1 + (-4.21 - 3.35i)T \)
good3 \( 1 + (2.19 + 1.74i)T + (0.667 + 2.92i)T^{2} \)
5 \( 1 + (-1.63 - 0.788i)T + (3.11 + 3.90i)T^{2} \)
7 \( 1 + (-0.882 + 1.10i)T + (-1.55 - 6.82i)T^{2} \)
11 \( 1 + (-3.44 - 0.787i)T + (9.91 + 4.77i)T^{2} \)
13 \( 1 + (1.23 - 5.42i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + 7.46iT - 17T^{2} \)
19 \( 1 + (3.93 - 3.14i)T + (4.22 - 18.5i)T^{2} \)
23 \( 1 + (1.61 - 0.776i)T + (14.3 - 17.9i)T^{2} \)
31 \( 1 + (2.07 - 4.31i)T + (-19.3 - 24.2i)T^{2} \)
37 \( 1 + (-1.04 + 0.239i)T + (33.3 - 16.0i)T^{2} \)
41 \( 1 + 1.71iT - 41T^{2} \)
43 \( 1 + (0.881 + 1.83i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (0.377 + 0.0861i)T + (42.3 + 20.3i)T^{2} \)
53 \( 1 + (-2.42 - 1.16i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + 3.81T + 59T^{2} \)
61 \( 1 + (10.5 + 8.42i)T + (13.5 + 59.4i)T^{2} \)
67 \( 1 + (0.659 + 2.88i)T + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (-1.39 + 6.12i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (0.416 + 0.865i)T + (-45.5 + 57.0i)T^{2} \)
79 \( 1 + (-2.71 + 0.619i)T + (71.1 - 34.2i)T^{2} \)
83 \( 1 + (5.65 + 7.09i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-2.30 + 4.78i)T + (-55.4 - 69.5i)T^{2} \)
97 \( 1 + (8.59 - 6.85i)T + (21.5 - 94.5i)T^{2} \)
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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

−14.23615872316310021455507670338, −13.83455176956665692981565562508, −12.30820932630870544481691694358, −11.74740596264248395767267801752, −10.69670163050143776264746367694, −9.367566073741005633597594899184, −7.09101510986522851716412729655, −6.24052614508980513058539960311, −4.68089385682342042289823770543, −1.76119050212757452755589190924, 4.23584119082040240358305104068, 5.56218829907150275546470909713, 6.22067447903633654560923758838, 8.439847036859778293009553321505, 9.783942793843136963485200045901, 10.85070628746394505524980425421, 12.12947563890291144011867343205, 13.16831424702990760392973978522, 14.86567233827377008233525846397, 15.37732720944824147161681618134

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