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Properties

Label 2-177-59.58-c10-0-13
Degree $2$
Conductor $177$
Sign $0.515 + 0.857i$
Analytic cond. $112.458$
Root an. cond. $10.6046$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 48.0i·2-s − 140.·3-s − 1.28e3·4-s − 1.21e3·5-s + 6.74e3i·6-s − 4.24e3·7-s + 1.26e4i·8-s + 1.96e4·9-s + 5.82e4i·10-s − 3.39e4i·11-s + 1.80e5·12-s − 4.13e5i·13-s + 2.03e5i·14-s + 1.70e5·15-s − 7.10e5·16-s − 2.03e6·17-s + ⋯
L(s)  = 1  − 1.50i·2-s − 0.577·3-s − 1.25·4-s − 0.388·5-s + 0.867i·6-s − 0.252·7-s + 0.384i·8-s + 0.333·9-s + 0.582i·10-s − 0.210i·11-s + 0.725·12-s − 1.11i·13-s + 0.378i·14-s + 0.224·15-s − 0.678·16-s − 1.43·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 + 0.857i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.515 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.515 + 0.857i$
Analytic conductor: \(112.458\)
Root analytic conductor: \(10.6046\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :5),\ 0.515 + 0.857i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.4977827760\)
\(L(\frac12)\) \(\approx\) \(0.4977827760\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 140.T \)
59 \( 1 + (3.68e8 + 6.12e8i)T \)
good2 \( 1 + 48.0iT - 1.02e3T^{2} \)
5 \( 1 + 1.21e3T + 9.76e6T^{2} \)
7 \( 1 + 4.24e3T + 2.82e8T^{2} \)
11 \( 1 + 3.39e4iT - 2.59e10T^{2} \)
13 \( 1 + 4.13e5iT - 1.37e11T^{2} \)
17 \( 1 + 2.03e6T + 2.01e12T^{2} \)
19 \( 1 - 9.46e5T + 6.13e12T^{2} \)
23 \( 1 - 2.44e6iT - 4.14e13T^{2} \)
29 \( 1 + 1.24e7T + 4.20e14T^{2} \)
31 \( 1 - 3.60e7iT - 8.19e14T^{2} \)
37 \( 1 + 1.04e8iT - 4.80e15T^{2} \)
41 \( 1 - 2.63e7T + 1.34e16T^{2} \)
43 \( 1 - 1.45e8iT - 2.16e16T^{2} \)
47 \( 1 - 2.23e8iT - 5.25e16T^{2} \)
53 \( 1 + 3.03e8T + 1.74e17T^{2} \)
61 \( 1 + 1.36e9iT - 7.13e17T^{2} \)
67 \( 1 - 2.43e9iT - 1.82e18T^{2} \)
71 \( 1 - 2.64e9T + 3.25e18T^{2} \)
73 \( 1 + 8.53e8iT - 4.29e18T^{2} \)
79 \( 1 - 4.49e7T + 9.46e18T^{2} \)
83 \( 1 - 1.00e9iT - 1.55e19T^{2} \)
89 \( 1 + 3.83e9iT - 3.11e19T^{2} \)
97 \( 1 - 2.35e9iT - 7.37e19T^{2} \)
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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

−11.03798704122751250443961124775, −9.980069450922099717013521463508, −9.091587797194489865816618909899, −7.73746020061454689310119057925, −6.42030290912630559618038508587, −5.08905513379853786862928300650, −3.94046151926261735467007465655, −3.00215531091012162529785959927, −1.75978214236527475218655031731, −0.55732812446496798709935547914, 0.20045035029359093521837248706, 2.05421504287023936162611532971, 4.05397033431520641850110750035, 4.88170839248376285695950656784, 6.10781129780354025630173346182, 6.77472269052333017163309036171, 7.64684401187286680398269265488, 8.758519560614700375810075285361, 9.682407547544189972470108318516, 11.15877246696202680576282647274

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