Properties

Label 2-177-1.1-c7-0-61
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  + 19.6·2-s − 27·3-s + 258.·4-s − 436.·5-s − 530.·6-s + 956.·7-s + 2.56e3·8-s + 729·9-s − 8.58e3·10-s + 1.97e3·11-s − 6.97e3·12-s − 5.30e3·13-s + 1.88e4·14-s + 1.17e4·15-s + 1.72e4·16-s − 1.91e4·17-s + 1.43e4·18-s − 2.14e4·19-s − 1.12e5·20-s − 2.58e4·21-s + 3.87e4·22-s − 3.43e4·23-s − 6.91e4·24-s + 1.12e5·25-s − 1.04e5·26-s − 1.96e4·27-s + 2.47e5·28-s + ⋯
L(s)  = 1  + 1.73·2-s − 0.577·3-s + 2.01·4-s − 1.56·5-s − 1.00·6-s + 1.05·7-s + 1.76·8-s + 0.333·9-s − 2.71·10-s + 0.446·11-s − 1.16·12-s − 0.670·13-s + 1.83·14-s + 0.902·15-s + 1.05·16-s − 0.947·17-s + 0.579·18-s − 0.717·19-s − 3.15·20-s − 0.608·21-s + 0.776·22-s − 0.588·23-s − 1.02·24-s + 1.44·25-s − 1.16·26-s − 0.192·27-s + 2.12·28-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 \( 1 - 19.6T + 128T^{2} \)
5 \( 1 + 436.T + 7.81e4T^{2} \)
7 \( 1 - 956.T + 8.23e5T^{2} \)
11 \( 1 - 1.97e3T + 1.94e7T^{2} \)
13 \( 1 + 5.30e3T + 6.27e7T^{2} \)
17 \( 1 + 1.91e4T + 4.10e8T^{2} \)
19 \( 1 + 2.14e4T + 8.93e8T^{2} \)
23 \( 1 + 3.43e4T + 3.40e9T^{2} \)
29 \( 1 - 4.00e4T + 1.72e10T^{2} \)
31 \( 1 + 9.54e4T + 2.75e10T^{2} \)
37 \( 1 + 2.69e5T + 9.49e10T^{2} \)
41 \( 1 + 3.60e5T + 1.94e11T^{2} \)
43 \( 1 + 9.29e5T + 2.71e11T^{2} \)
47 \( 1 - 5.40e4T + 5.06e11T^{2} \)
53 \( 1 - 6.80e5T + 1.17e12T^{2} \)
61 \( 1 + 1.27e6T + 3.14e12T^{2} \)
67 \( 1 - 6.30e5T + 6.06e12T^{2} \)
71 \( 1 - 2.05e6T + 9.09e12T^{2} \)
73 \( 1 + 5.36e6T + 1.10e13T^{2} \)
79 \( 1 - 4.57e6T + 1.92e13T^{2} \)
83 \( 1 - 9.49e6T + 2.71e13T^{2} \)
89 \( 1 + 2.45e6T + 4.42e13T^{2} \)
97 \( 1 + 1.41e7T + 8.07e13T^{2} \)
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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

−11.49196865990953424151177364460, −10.61186261529811497687923605761, −8.492516560015961289087758517129, −7.37364678210501767160905896615, −6.51274905947508546758278277448, −5.04918757563897706118054794272, −4.45694433219675826927404614465, −3.57323160214835986380401936263, −1.93410422473796113134792743784, 0, 1.93410422473796113134792743784, 3.57323160214835986380401936263, 4.45694433219675826927404614465, 5.04918757563897706118054794272, 6.51274905947508546758278277448, 7.37364678210501767160905896615, 8.492516560015961289087758517129, 10.61186261529811497687923605761, 11.49196865990953424151177364460

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