Properties

Label 2-177-1.1-c7-0-15
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $55.2921$
Root an. cond. $7.43586$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 14.3·2-s − 27·3-s + 76.6·4-s + 348.·5-s + 386.·6-s − 1.31e3·7-s + 734.·8-s + 729·9-s − 4.99e3·10-s + 4.99e3·11-s − 2.06e3·12-s + 3.39e3·13-s + 1.88e4·14-s − 9.41e3·15-s − 2.03e4·16-s + 1.30e4·17-s − 1.04e4·18-s + 8.79e3·19-s + 2.67e4·20-s + 3.55e4·21-s − 7.14e4·22-s + 1.07e5·23-s − 1.98e4·24-s + 4.35e4·25-s − 4.86e4·26-s − 1.96e4·27-s − 1.00e5·28-s + ⋯
L(s)  = 1  − 1.26·2-s − 0.577·3-s + 0.598·4-s + 1.24·5-s + 0.729·6-s − 1.45·7-s + 0.507·8-s + 0.333·9-s − 1.57·10-s + 1.13·11-s − 0.345·12-s + 0.428·13-s + 1.83·14-s − 0.720·15-s − 1.24·16-s + 0.645·17-s − 0.421·18-s + 0.294·19-s + 0.747·20-s + 0.837·21-s − 1.43·22-s + 1.84·23-s − 0.293·24-s + 0.558·25-s − 0.542·26-s − 0.192·27-s − 0.868·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(55.2921\)
Root analytic conductor: \(7.43586\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(0.9510276257\)
\(L(\frac12)\) \(\approx\) \(0.9510276257\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 27T \)
59 \( 1 + 2.05e5T \)
good2 \( 1 + 14.3T + 128T^{2} \)
5 \( 1 - 348.T + 7.81e4T^{2} \)
7 \( 1 + 1.31e3T + 8.23e5T^{2} \)
11 \( 1 - 4.99e3T + 1.94e7T^{2} \)
13 \( 1 - 3.39e3T + 6.27e7T^{2} \)
17 \( 1 - 1.30e4T + 4.10e8T^{2} \)
19 \( 1 - 8.79e3T + 8.93e8T^{2} \)
23 \( 1 - 1.07e5T + 3.40e9T^{2} \)
29 \( 1 - 3.19e4T + 1.72e10T^{2} \)
31 \( 1 + 1.54e5T + 2.75e10T^{2} \)
37 \( 1 + 5.28e5T + 9.49e10T^{2} \)
41 \( 1 + 4.80e5T + 1.94e11T^{2} \)
43 \( 1 + 4.38e5T + 2.71e11T^{2} \)
47 \( 1 - 1.23e6T + 5.06e11T^{2} \)
53 \( 1 + 1.18e6T + 1.17e12T^{2} \)
61 \( 1 - 8.73e5T + 3.14e12T^{2} \)
67 \( 1 + 8.32e5T + 6.06e12T^{2} \)
71 \( 1 - 3.15e6T + 9.09e12T^{2} \)
73 \( 1 - 2.69e6T + 1.10e13T^{2} \)
79 \( 1 - 6.93e6T + 1.92e13T^{2} \)
83 \( 1 + 3.14e6T + 2.71e13T^{2} \)
89 \( 1 + 4.70e5T + 4.42e13T^{2} \)
97 \( 1 - 1.03e7T + 8.07e13T^{2} \)
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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

−10.96092194504438051138010976746, −10.09756489850938137502089121669, −9.434388340968397339589141910600, −8.844370439997100279816956557061, −7.06090833758286326314568147375, −6.45371587881169298184934145803, −5.28123827223469590747603875338, −3.42994767035253593077028118899, −1.68464473817070593883085396031, −0.70334622483954518117393701212, 0.70334622483954518117393701212, 1.68464473817070593883085396031, 3.42994767035253593077028118899, 5.28123827223469590747603875338, 6.45371587881169298184934145803, 7.06090833758286326314568147375, 8.844370439997100279816956557061, 9.434388340968397339589141910600, 10.09756489850938137502089121669, 10.96092194504438051138010976746

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