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Properties

Label 2-177-59.58-c10-0-28
Degree $2$
Conductor $177$
Sign $-0.441 + 0.897i$
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  − 45.0i·2-s + 140.·3-s − 1.00e3·4-s − 5.62e3·5-s − 6.31e3i·6-s − 2.21e4·7-s − 897. i·8-s + 1.96e4·9-s + 2.53e5i·10-s − 2.86e4i·11-s − 1.40e5·12-s + 4.05e5i·13-s + 9.99e5i·14-s − 7.89e5·15-s − 1.06e6·16-s + 1.93e5·17-s + ⋯
L(s)  = 1  − 1.40i·2-s + 0.577·3-s − 0.980·4-s − 1.80·5-s − 0.812i·6-s − 1.32·7-s − 0.0273i·8-s + 0.333·9-s + 2.53i·10-s − 0.178i·11-s − 0.566·12-s + 1.09i·13-s + 1.85i·14-s − 1.03·15-s − 1.01·16-s + 0.136·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.441 + 0.897i$
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: \((2,\ 177,\ (\ :5),\ -0.441 + 0.897i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.6284742898\)
\(L(\frac12)\) \(\approx\) \(0.6284742898\)
\(L(6)\) 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 \( 1 + (3.15e8 - 6.41e8i)T \)
good2 \( 1 + 45.0iT - 1.02e3T^{2} \)
5 \( 1 + 5.62e3T + 9.76e6T^{2} \)
7 \( 1 + 2.21e4T + 2.82e8T^{2} \)
11 \( 1 + 2.86e4iT - 2.59e10T^{2} \)
13 \( 1 - 4.05e5iT - 1.37e11T^{2} \)
17 \( 1 - 1.93e5T + 2.01e12T^{2} \)
19 \( 1 + 1.39e6T + 6.13e12T^{2} \)
23 \( 1 - 9.59e6iT - 4.14e13T^{2} \)
29 \( 1 + 8.13e6T + 4.20e14T^{2} \)
31 \( 1 - 8.53e6iT - 8.19e14T^{2} \)
37 \( 1 + 3.75e7iT - 4.80e15T^{2} \)
41 \( 1 + 1.74e8T + 1.34e16T^{2} \)
43 \( 1 + 1.52e8iT - 2.16e16T^{2} \)
47 \( 1 - 2.46e8iT - 5.25e16T^{2} \)
53 \( 1 + 7.65e8T + 1.74e17T^{2} \)
61 \( 1 - 2.67e8iT - 7.13e17T^{2} \)
67 \( 1 + 1.82e9iT - 1.82e18T^{2} \)
71 \( 1 - 3.31e9T + 3.25e18T^{2} \)
73 \( 1 + 3.54e9iT - 4.29e18T^{2} \)
79 \( 1 + 4.35e9T + 9.46e18T^{2} \)
83 \( 1 + 2.12e9iT - 1.55e19T^{2} \)
89 \( 1 + 2.36e9iT - 3.11e19T^{2} \)
97 \( 1 + 1.02e9iT - 7.37e19T^{2} \)
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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

−10.72147405993953821397419350711, −9.565567113412618848583948451289, −8.857265056633466574811490929063, −7.56399611899688850507144439420, −6.64337490981157289362947100424, −4.45910586676055078720247322004, −3.55102050398526966834698011051, −3.18498846865015509916400122265, −1.71263705559333442419760030404, −0.31333177283866594453093209359, 0.39060000536487266290138705867, 2.82896446663898528558011533925, 3.77528232385006945598893853399, 4.87656429386284600526090730642, 6.40120855464742428175246068171, 7.09854218443970258819896836366, 8.099937161944487246203933423324, 8.511661264464657712860142463903, 9.889155639122598786724788344220, 11.14284800415870284844268713602

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