Properties

Label 2-177-1.1-c7-0-0
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  − 3.88·2-s − 27·3-s − 112.·4-s − 26.2·5-s + 104.·6-s − 644.·7-s + 936.·8-s + 729·9-s + 102.·10-s − 5.69e3·11-s + 3.04e3·12-s − 606.·13-s + 2.50e3·14-s + 709.·15-s + 1.08e4·16-s − 3.92e4·17-s − 2.83e3·18-s − 3.40e4·19-s + 2.96e3·20-s + 1.74e4·21-s + 2.21e4·22-s − 2.12e4·23-s − 2.52e4·24-s − 7.74e4·25-s + 2.35e3·26-s − 1.96e4·27-s + 7.27e4·28-s + ⋯
L(s)  = 1  − 0.343·2-s − 0.577·3-s − 0.882·4-s − 0.0939·5-s + 0.198·6-s − 0.710·7-s + 0.646·8-s + 0.333·9-s + 0.0322·10-s − 1.28·11-s + 0.509·12-s − 0.0765·13-s + 0.243·14-s + 0.0542·15-s + 0.659·16-s − 1.93·17-s − 0.114·18-s − 1.13·19-s + 0.0828·20-s + 0.410·21-s + 0.442·22-s − 0.364·23-s − 0.373·24-s − 0.991·25-s + 0.0262·26-s − 0.192·27-s + 0.626·28-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

\(L(4)\) \(\approx\) \(0.007204235492\)
\(L(\frac12)\) \(\approx\) \(0.007204235492\)
\(L(\frac{9}{2})\) 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 \( 1 + 3.88T + 128T^{2} \)
5 \( 1 + 26.2T + 7.81e4T^{2} \)
7 \( 1 + 644.T + 8.23e5T^{2} \)
11 \( 1 + 5.69e3T + 1.94e7T^{2} \)
13 \( 1 + 606.T + 6.27e7T^{2} \)
17 \( 1 + 3.92e4T + 4.10e8T^{2} \)
19 \( 1 + 3.40e4T + 8.93e8T^{2} \)
23 \( 1 + 2.12e4T + 3.40e9T^{2} \)
29 \( 1 - 2.61e4T + 1.72e10T^{2} \)
31 \( 1 + 3.10e5T + 2.75e10T^{2} \)
37 \( 1 + 3.30e5T + 9.49e10T^{2} \)
41 \( 1 - 3.18e5T + 1.94e11T^{2} \)
43 \( 1 - 1.67e4T + 2.71e11T^{2} \)
47 \( 1 - 7.64e5T + 5.06e11T^{2} \)
53 \( 1 + 1.38e6T + 1.17e12T^{2} \)
61 \( 1 + 3.54e5T + 3.14e12T^{2} \)
67 \( 1 - 4.54e6T + 6.06e12T^{2} \)
71 \( 1 + 3.67e5T + 9.09e12T^{2} \)
73 \( 1 + 1.13e6T + 1.10e13T^{2} \)
79 \( 1 - 3.04e6T + 1.92e13T^{2} \)
83 \( 1 + 8.39e6T + 2.71e13T^{2} \)
89 \( 1 + 3.28e6T + 4.42e13T^{2} \)
97 \( 1 - 8.46e6T + 8.07e13T^{2} \)
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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

−11.10229280246720388088614293380, −10.41674829570032797699981070828, −9.402130789358016481188734778215, −8.468442286917397274098503711489, −7.31218007745734562015194751431, −6.06780053545432742656865232633, −4.91924307623674576190623502250, −3.88404465633245262407654514559, −2.11051710880023140051078302191, −0.04779777867369096037625480846, 0.04779777867369096037625480846, 2.11051710880023140051078302191, 3.88404465633245262407654514559, 4.91924307623674576190623502250, 6.06780053545432742656865232633, 7.31218007745734562015194751431, 8.468442286917397274098503711489, 9.402130789358016481188734778215, 10.41674829570032797699981070828, 11.10229280246720388088614293380

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