Properties

Label 2-177-1.1-c7-0-23
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  − 21.6·2-s + 27·3-s + 340.·4-s − 399.·5-s − 584.·6-s − 1.78e3·7-s − 4.59e3·8-s + 729·9-s + 8.63e3·10-s − 204.·11-s + 9.18e3·12-s + 7.46e3·13-s + 3.85e4·14-s − 1.07e4·15-s + 5.58e4·16-s + 2.07e3·17-s − 1.57e4·18-s + 3.86e4·19-s − 1.35e5·20-s − 4.80e4·21-s + 4.41e3·22-s − 4.76e4·23-s − 1.23e5·24-s + 8.11e4·25-s − 1.61e5·26-s + 1.96e4·27-s − 6.05e5·28-s + ⋯
L(s)  = 1  − 1.91·2-s + 0.577·3-s + 2.65·4-s − 1.42·5-s − 1.10·6-s − 1.96·7-s − 3.17·8-s + 0.333·9-s + 2.73·10-s − 0.0462·11-s + 1.53·12-s + 0.942·13-s + 3.75·14-s − 0.824·15-s + 3.40·16-s + 0.102·17-s − 0.637·18-s + 1.29·19-s − 3.79·20-s − 1.13·21-s + 0.0884·22-s − 0.815·23-s − 1.83·24-s + 1.03·25-s − 1.80·26-s + 0.192·27-s − 5.21·28-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 \( 1 + 21.6T + 128T^{2} \)
5 \( 1 + 399.T + 7.81e4T^{2} \)
7 \( 1 + 1.78e3T + 8.23e5T^{2} \)
11 \( 1 + 204.T + 1.94e7T^{2} \)
13 \( 1 - 7.46e3T + 6.27e7T^{2} \)
17 \( 1 - 2.07e3T + 4.10e8T^{2} \)
19 \( 1 - 3.86e4T + 8.93e8T^{2} \)
23 \( 1 + 4.76e4T + 3.40e9T^{2} \)
29 \( 1 - 2.23e5T + 1.72e10T^{2} \)
31 \( 1 - 2.73e5T + 2.75e10T^{2} \)
37 \( 1 + 5.45e5T + 9.49e10T^{2} \)
41 \( 1 + 5.21e5T + 1.94e11T^{2} \)
43 \( 1 - 3.27e5T + 2.71e11T^{2} \)
47 \( 1 + 2.41e4T + 5.06e11T^{2} \)
53 \( 1 + 1.06e6T + 1.17e12T^{2} \)
61 \( 1 - 1.08e6T + 3.14e12T^{2} \)
67 \( 1 + 5.57e4T + 6.06e12T^{2} \)
71 \( 1 + 8.51e5T + 9.09e12T^{2} \)
73 \( 1 - 3.44e6T + 1.10e13T^{2} \)
79 \( 1 + 8.05e6T + 1.92e13T^{2} \)
83 \( 1 - 2.05e6T + 2.71e13T^{2} \)
89 \( 1 + 1.68e6T + 4.42e13T^{2} \)
97 \( 1 - 1.14e7T + 8.07e13T^{2} \)
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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

−10.41031703776011911992093375974, −9.812640651268206000097041314654, −8.767083050427901914979773454205, −8.097873996556362929220184772324, −7.08766895895843462790893350639, −6.34675214676509276874845988712, −3.55919338402245537245530401928, −2.89704787934990501324148658654, −0.981804842422620565561853816225, 0, 0.981804842422620565561853816225, 2.89704787934990501324148658654, 3.55919338402245537245530401928, 6.34675214676509276874845988712, 7.08766895895843462790893350639, 8.097873996556362929220184772324, 8.767083050427901914979773454205, 9.812640651268206000097041314654, 10.41031703776011911992093375974

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