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Properties

Label 2-59-59.58-c6-0-13
Degree $2$
Conductor $59$
Sign $-0.423 + 0.905i$
Analytic cond. $13.5731$
Root an. cond. $3.68418$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 11.1i·2-s − 46.4·3-s − 59.3·4-s + 157.·5-s + 516. i·6-s + 407.·7-s − 51.1i·8-s + 1.43e3·9-s − 1.75e3i·10-s + 2.36e3i·11-s + 2.76e3·12-s − 1.83e3i·13-s − 4.53e3i·14-s − 7.34e3·15-s − 4.36e3·16-s + 1.70e3·17-s + ⋯
L(s)  = 1  − 1.38i·2-s − 1.72·3-s − 0.928·4-s + 1.26·5-s + 2.39i·6-s + 1.18·7-s − 0.0999i·8-s + 1.96·9-s − 1.75i·10-s + 1.77i·11-s + 1.59·12-s − 0.835i·13-s − 1.65i·14-s − 2.17·15-s − 1.06·16-s + 0.347·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.423 + 0.905i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.423 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(59\)
Sign: $-0.423 + 0.905i$
Analytic conductor: \(13.5731\)
Root analytic conductor: \(3.68418\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 59,\ (\ :3),\ -0.423 + 0.905i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.797907 - 1.25451i\)
\(L(\frac12)\) \(\approx\) \(0.797907 - 1.25451i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 + (-8.70e4 + 1.86e5i)T \)
good2 \( 1 + 11.1iT - 64T^{2} \)
3 \( 1 + 46.4T + 729T^{2} \)
5 \( 1 - 157.T + 1.56e4T^{2} \)
7 \( 1 - 407.T + 1.17e5T^{2} \)
11 \( 1 - 2.36e3iT - 1.77e6T^{2} \)
13 \( 1 + 1.83e3iT - 4.82e6T^{2} \)
17 \( 1 - 1.70e3T + 2.41e7T^{2} \)
19 \( 1 - 1.01e4T + 4.70e7T^{2} \)
23 \( 1 + 1.16e4iT - 1.48e8T^{2} \)
29 \( 1 - 5.12e3T + 5.94e8T^{2} \)
31 \( 1 - 1.89e4iT - 8.87e8T^{2} \)
37 \( 1 + 9.20e4iT - 2.56e9T^{2} \)
41 \( 1 - 7.36e4T + 4.75e9T^{2} \)
43 \( 1 - 5.20e3iT - 6.32e9T^{2} \)
47 \( 1 + 7.90e4iT - 1.07e10T^{2} \)
53 \( 1 + 1.32e5T + 2.21e10T^{2} \)
61 \( 1 - 2.06e5iT - 5.15e10T^{2} \)
67 \( 1 - 3.66e5iT - 9.04e10T^{2} \)
71 \( 1 - 6.82e5T + 1.28e11T^{2} \)
73 \( 1 - 1.32e4iT - 1.51e11T^{2} \)
79 \( 1 + 4.18e5T + 2.43e11T^{2} \)
83 \( 1 - 5.57e4iT - 3.26e11T^{2} \)
89 \( 1 + 1.20e6iT - 4.96e11T^{2} \)
97 \( 1 + 1.41e5iT - 8.32e11T^{2} \)
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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

−12.82439041886556220669546059543, −12.24478162939522818138915615489, −11.21261679745340911140290230017, −10.32550493445283692085184671623, −9.687385165576239902356227464997, −7.16255698612963749523875609218, −5.54869536452045317509234837978, −4.64712903698493702752979245743, −2.03755499453992696242291846095, −0.985037163490887919361031882775, 1.23052924848523117004542016202, 5.00971882997659305124709334685, 5.69041583533323128071158577049, 6.43123533069457545046283511626, 7.88086612476645624934462227265, 9.488991151789635106375013558955, 11.09181233466197071692612212874, 11.63988912624347906370839132513, 13.62029181168321534776860756931, 14.15715574736590079574872061008

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