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Properties

Label 2-177-59.58-c8-0-36
Degree $2$
Conductor $177$
Sign $-0.303 + 0.952i$
Analytic cond. $72.1060$
Root an. cond. $8.49152$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 25.6i·2-s + 46.7·3-s − 400.·4-s − 239.·5-s − 1.19e3i·6-s − 1.75e3·7-s + 3.71e3i·8-s + 2.18e3·9-s + 6.14e3i·10-s + 3.30e3i·11-s − 1.87e4·12-s + 1.31e4i·13-s + 4.49e4i·14-s − 1.12e4·15-s − 7.47e3·16-s + 1.19e5·17-s + ⋯
L(s)  = 1  − 1.60i·2-s + 0.577·3-s − 1.56·4-s − 0.383·5-s − 0.924i·6-s − 0.731·7-s + 0.906i·8-s + 0.333·9-s + 0.614i·10-s + 0.225i·11-s − 0.904·12-s + 0.460i·13-s + 1.17i·14-s − 0.221·15-s − 0.114·16-s + 1.43·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.303 + 0.952i$
Analytic conductor: \(72.1060\)
Root analytic conductor: \(8.49152\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :4),\ -0.303 + 0.952i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.044249476\)
\(L(\frac12)\) \(\approx\) \(2.044249476\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 46.7T \)
59 \( 1 + (-3.67e6 + 1.15e7i)T \)
good2 \( 1 + 25.6iT - 256T^{2} \)
5 \( 1 + 239.T + 3.90e5T^{2} \)
7 \( 1 + 1.75e3T + 5.76e6T^{2} \)
11 \( 1 - 3.30e3iT - 2.14e8T^{2} \)
13 \( 1 - 1.31e4iT - 8.15e8T^{2} \)
17 \( 1 - 1.19e5T + 6.97e9T^{2} \)
19 \( 1 - 1.48e5T + 1.69e10T^{2} \)
23 \( 1 - 1.44e5iT - 7.83e10T^{2} \)
29 \( 1 - 3.97e5T + 5.00e11T^{2} \)
31 \( 1 - 1.82e6iT - 8.52e11T^{2} \)
37 \( 1 + 2.78e5iT - 3.51e12T^{2} \)
41 \( 1 - 1.92e6T + 7.98e12T^{2} \)
43 \( 1 + 3.71e6iT - 1.16e13T^{2} \)
47 \( 1 + 4.04e6iT - 2.38e13T^{2} \)
53 \( 1 + 1.28e6T + 6.22e13T^{2} \)
61 \( 1 + 2.00e7iT - 1.91e14T^{2} \)
67 \( 1 + 6.13e6iT - 4.06e14T^{2} \)
71 \( 1 - 3.64e7T + 6.45e14T^{2} \)
73 \( 1 + 1.88e7iT - 8.06e14T^{2} \)
79 \( 1 - 7.26e7T + 1.51e15T^{2} \)
83 \( 1 - 5.12e7iT - 2.25e15T^{2} \)
89 \( 1 - 4.51e7iT - 3.93e15T^{2} \)
97 \( 1 - 6.88e7iT - 7.83e15T^{2} \)
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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

−10.91374932315436106692621196013, −9.839461313361037996612377730440, −9.429077135372978755969978908863, −8.126987052658911409424866220851, −6.91026980049088725882777012512, −5.12160187851682340631935996197, −3.65680425837070707076820521277, −3.21855922321326431892512385379, −1.87313219464716316138978504654, −0.75019835306580224924804365036, 0.70550069575377441053345618847, 2.89536717159454265566700586846, 4.09296328489758290676023448665, 5.48342509410544161738576831048, 6.34542342493150177515017350920, 7.65500868128644462909169885264, 7.931835136697548362590768085624, 9.247540213809073647144353466496, 9.988046890001689683042510008599, 11.63457938925690865272909895406

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