Properties

Label 2-177-1.1-c9-0-33
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $91.1613$
Root an. cond. $9.54784$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 31.5·2-s − 81·3-s + 483.·4-s + 1.95e3·5-s − 2.55e3·6-s − 4.90e3·7-s − 888.·8-s + 6.56e3·9-s + 6.17e4·10-s − 7.90e4·11-s − 3.91e4·12-s − 1.84e3·13-s − 1.54e5·14-s − 1.58e5·15-s − 2.75e5·16-s + 4.59e5·17-s + 2.07e5·18-s + 1.08e6·19-s + 9.46e5·20-s + 3.97e5·21-s − 2.49e6·22-s + 2.20e6·23-s + 7.20e4·24-s + 1.87e6·25-s − 5.82e4·26-s − 5.31e5·27-s − 2.37e6·28-s + ⋯
L(s)  = 1  + 1.39·2-s − 0.577·3-s + 0.944·4-s + 1.39·5-s − 0.805·6-s − 0.771·7-s − 0.0767·8-s + 0.333·9-s + 1.95·10-s − 1.62·11-s − 0.545·12-s − 0.0179·13-s − 1.07·14-s − 0.808·15-s − 1.05·16-s + 1.33·17-s + 0.464·18-s + 1.90·19-s + 1.32·20-s + 0.445·21-s − 2.27·22-s + 1.64·23-s + 0.0442·24-s + 0.959·25-s − 0.0250·26-s − 0.192·27-s − 0.729·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(91.1613\)
Root analytic conductor: \(9.54784\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(4.436602954\)
\(L(\frac12)\) \(\approx\) \(4.436602954\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 81T \)
59 \( 1 - 1.21e7T \)
good2 \( 1 - 31.5T + 512T^{2} \)
5 \( 1 - 1.95e3T + 1.95e6T^{2} \)
7 \( 1 + 4.90e3T + 4.03e7T^{2} \)
11 \( 1 + 7.90e4T + 2.35e9T^{2} \)
13 \( 1 + 1.84e3T + 1.06e10T^{2} \)
17 \( 1 - 4.59e5T + 1.18e11T^{2} \)
19 \( 1 - 1.08e6T + 3.22e11T^{2} \)
23 \( 1 - 2.20e6T + 1.80e12T^{2} \)
29 \( 1 + 4.56e6T + 1.45e13T^{2} \)
31 \( 1 - 6.82e6T + 2.64e13T^{2} \)
37 \( 1 - 1.05e7T + 1.29e14T^{2} \)
41 \( 1 + 1.66e7T + 3.27e14T^{2} \)
43 \( 1 - 3.40e7T + 5.02e14T^{2} \)
47 \( 1 - 3.70e6T + 1.11e15T^{2} \)
53 \( 1 + 2.35e7T + 3.29e15T^{2} \)
61 \( 1 - 1.59e8T + 1.16e16T^{2} \)
67 \( 1 - 1.14e8T + 2.72e16T^{2} \)
71 \( 1 - 1.06e8T + 4.58e16T^{2} \)
73 \( 1 - 2.12e7T + 5.88e16T^{2} \)
79 \( 1 - 3.40e8T + 1.19e17T^{2} \)
83 \( 1 + 9.07e7T + 1.86e17T^{2} \)
89 \( 1 - 7.56e8T + 3.50e17T^{2} \)
97 \( 1 - 3.71e8T + 7.60e17T^{2} \)
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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

−11.21363268615545266139964628143, −10.04115360703614605588320520381, −9.445044210550205354662061085262, −7.51267438579237762228247911325, −6.33000172879414084355269900458, −5.37379265227051123769857395359, −5.18776386131649470197859621603, −3.31112268784197654499451265893, −2.54698173657369464280833562063, −0.871960846110275261303596227028, 0.871960846110275261303596227028, 2.54698173657369464280833562063, 3.31112268784197654499451265893, 5.18776386131649470197859621603, 5.37379265227051123769857395359, 6.33000172879414084355269900458, 7.51267438579237762228247911325, 9.445044210550205354662061085262, 10.04115360703614605588320520381, 11.21363268615545266139964628143

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