Properties

Label 2-177-1.1-c7-0-37
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.01·2-s + 27·3-s − 91.8·4-s − 385.·5-s − 162.·6-s + 847.·7-s + 1.32e3·8-s + 729·9-s + 2.32e3·10-s − 5.52e3·11-s − 2.47e3·12-s + 2.98e3·13-s − 5.09e3·14-s − 1.04e4·15-s + 3.79e3·16-s − 6.65e3·17-s − 4.38e3·18-s + 4.74e4·19-s + 3.54e4·20-s + 2.28e4·21-s + 3.32e4·22-s + 1.69e4·23-s + 3.56e4·24-s + 7.07e4·25-s − 1.79e4·26-s + 1.96e4·27-s − 7.78e4·28-s + ⋯
L(s)  = 1  − 0.531·2-s + 0.577·3-s − 0.717·4-s − 1.38·5-s − 0.306·6-s + 0.934·7-s + 0.913·8-s + 0.333·9-s + 0.733·10-s − 1.25·11-s − 0.414·12-s + 0.377·13-s − 0.496·14-s − 0.796·15-s + 0.231·16-s − 0.328·17-s − 0.177·18-s + 1.58·19-s + 0.990·20-s + 0.539·21-s + 0.665·22-s + 0.289·23-s + 0.527·24-s + 0.905·25-s − 0.200·26-s + 0.192·27-s − 0.670·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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 \( 1 + 6.01T + 128T^{2} \)
5 \( 1 + 385.T + 7.81e4T^{2} \)
7 \( 1 - 847.T + 8.23e5T^{2} \)
11 \( 1 + 5.52e3T + 1.94e7T^{2} \)
13 \( 1 - 2.98e3T + 6.27e7T^{2} \)
17 \( 1 + 6.65e3T + 4.10e8T^{2} \)
19 \( 1 - 4.74e4T + 8.93e8T^{2} \)
23 \( 1 - 1.69e4T + 3.40e9T^{2} \)
29 \( 1 - 2.63e3T + 1.72e10T^{2} \)
31 \( 1 - 1.38e5T + 2.75e10T^{2} \)
37 \( 1 + 9.12e4T + 9.49e10T^{2} \)
41 \( 1 - 5.49e5T + 1.94e11T^{2} \)
43 \( 1 + 2.40e5T + 2.71e11T^{2} \)
47 \( 1 + 1.17e6T + 5.06e11T^{2} \)
53 \( 1 + 1.00e5T + 1.17e12T^{2} \)
61 \( 1 - 7.14e4T + 3.14e12T^{2} \)
67 \( 1 + 9.12e5T + 6.06e12T^{2} \)
71 \( 1 + 2.74e5T + 9.09e12T^{2} \)
73 \( 1 + 5.34e6T + 1.10e13T^{2} \)
79 \( 1 - 1.12e6T + 1.92e13T^{2} \)
83 \( 1 + 4.95e6T + 2.71e13T^{2} \)
89 \( 1 + 8.64e6T + 4.42e13T^{2} \)
97 \( 1 + 2.69e6T + 8.07e13T^{2} \)
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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

−10.86364300025879140415040856714, −9.763141735804097828029703165848, −8.558759158687989768346789012548, −7.977088681158857864264839654020, −7.39442906986920544843810609332, −5.15626665856260100645956724457, −4.30683664316772472043747324407, −3.08055702574691546666819277131, −1.25359279216617564270454363540, 0, 1.25359279216617564270454363540, 3.08055702574691546666819277131, 4.30683664316772472043747324407, 5.15626665856260100645956724457, 7.39442906986920544843810609332, 7.977088681158857864264839654020, 8.558759158687989768346789012548, 9.763141735804097828029703165848, 10.86364300025879140415040856714

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