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Properties

Label 2-177-177.176-c5-0-57
Degree $2$
Conductor $177$
Sign $0.247 + 0.968i$
Analytic cond. $28.3879$
Root an. cond. $5.32803$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.56·2-s + (11.2 − 10.8i)3-s + 25.2·4-s + 51.3i·5-s + (−84.9 + 81.7i)6-s + 90.7·7-s + 51.4·8-s + (9.51 − 242. i)9-s − 388. i·10-s − 682.·11-s + (283. − 272. i)12-s − 90.5i·13-s − 686.·14-s + (554. + 577. i)15-s − 1.19e3·16-s − 771. i·17-s + ⋯
L(s)  = 1  − 1.33·2-s + (0.720 − 0.693i)3-s + 0.787·4-s + 0.918i·5-s + (−0.963 + 0.926i)6-s + 0.699·7-s + 0.284·8-s + (0.0391 − 0.999i)9-s − 1.22i·10-s − 1.70·11-s + (0.567 − 0.545i)12-s − 0.148i·13-s − 0.935·14-s + (0.636 + 0.662i)15-s − 1.16·16-s − 0.647i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.247 + 0.968i$
Analytic conductor: \(28.3879\)
Root analytic conductor: \(5.32803\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ 0.247 + 0.968i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.053104253\)
\(L(\frac12)\) \(\approx\) \(1.053104253\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-11.2 + 10.8i)T \)
59 \( 1 + (-1.31e4 + 2.32e4i)T \)
good2 \( 1 + 7.56T + 32T^{2} \)
5 \( 1 - 51.3iT - 3.12e3T^{2} \)
7 \( 1 - 90.7T + 1.68e4T^{2} \)
11 \( 1 + 682.T + 1.61e5T^{2} \)
13 \( 1 + 90.5iT - 3.71e5T^{2} \)
17 \( 1 + 771. iT - 1.41e6T^{2} \)
19 \( 1 - 2.48e3T + 2.47e6T^{2} \)
23 \( 1 + 440.T + 6.43e6T^{2} \)
29 \( 1 - 6.84e3iT - 2.05e7T^{2} \)
31 \( 1 - 1.05e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.04e4iT - 6.93e7T^{2} \)
41 \( 1 + 1.66e4iT - 1.15e8T^{2} \)
43 \( 1 - 6.37e3iT - 1.47e8T^{2} \)
47 \( 1 - 2.77e4T + 2.29e8T^{2} \)
53 \( 1 + 2.61e4iT - 4.18e8T^{2} \)
61 \( 1 - 9.40e3iT - 8.44e8T^{2} \)
67 \( 1 + 7.31e4iT - 1.35e9T^{2} \)
71 \( 1 - 4.37e4iT - 1.80e9T^{2} \)
73 \( 1 + 3.79e4iT - 2.07e9T^{2} \)
79 \( 1 + 1.86e4T + 3.07e9T^{2} \)
83 \( 1 - 1.08e5T + 3.93e9T^{2} \)
89 \( 1 + 5.17e4T + 5.58e9T^{2} \)
97 \( 1 + 9.09e4iT - 8.58e9T^{2} \)
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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

−11.18243670845930199158856991734, −10.47833291722055271150320503555, −9.443796358433020442572607626411, −8.422181750863777421435393516725, −7.50940715893873670326693510394, −7.14924555938972140641369286089, −5.26194056499091918938109084756, −3.10655896154665554929596946298, −2.02730735705466288981095792932, −0.56226647870909850331989513447, 1.11174025409102954011245798790, 2.52403404773710148776872289549, 4.43365400221347543416770315345, 5.33607778894581148912293016859, 7.69311210677242125059364335296, 8.053149468086841432206684990622, 8.937183083332067137263593118050, 9.865535865653923975386849474738, 10.53807737352007141483943979643, 11.63385687735179075041745331289

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