Properties

Label 2-177-1.1-c7-0-66
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  + 18.7·2-s − 27·3-s + 222.·4-s + 443.·5-s − 505.·6-s − 1.69e3·7-s + 1.76e3·8-s + 729·9-s + 8.30e3·10-s − 2.76e3·11-s − 6.00e3·12-s − 1.09e4·13-s − 3.17e4·14-s − 1.19e4·15-s + 4.61e3·16-s − 1.71e4·17-s + 1.36e4·18-s − 4.12e4·19-s + 9.87e4·20-s + 4.57e4·21-s − 5.17e4·22-s + 8.25e4·23-s − 4.77e4·24-s + 1.18e5·25-s − 2.05e5·26-s − 1.96e4·27-s − 3.77e5·28-s + ⋯
L(s)  = 1  + 1.65·2-s − 0.577·3-s + 1.73·4-s + 1.58·5-s − 0.955·6-s − 1.86·7-s + 1.22·8-s + 0.333·9-s + 2.62·10-s − 0.626·11-s − 1.00·12-s − 1.38·13-s − 3.09·14-s − 0.916·15-s + 0.281·16-s − 0.847·17-s + 0.551·18-s − 1.38·19-s + 2.75·20-s + 1.07·21-s − 1.03·22-s + 1.41·23-s − 0.704·24-s + 1.52·25-s − 2.28·26-s − 0.192·27-s − 3.24·28-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 \( 1 - 18.7T + 128T^{2} \)
5 \( 1 - 443.T + 7.81e4T^{2} \)
7 \( 1 + 1.69e3T + 8.23e5T^{2} \)
11 \( 1 + 2.76e3T + 1.94e7T^{2} \)
13 \( 1 + 1.09e4T + 6.27e7T^{2} \)
17 \( 1 + 1.71e4T + 4.10e8T^{2} \)
19 \( 1 + 4.12e4T + 8.93e8T^{2} \)
23 \( 1 - 8.25e4T + 3.40e9T^{2} \)
29 \( 1 - 1.03e5T + 1.72e10T^{2} \)
31 \( 1 + 1.62e5T + 2.75e10T^{2} \)
37 \( 1 - 4.46e5T + 9.49e10T^{2} \)
41 \( 1 - 3.71e5T + 1.94e11T^{2} \)
43 \( 1 + 7.90e5T + 2.71e11T^{2} \)
47 \( 1 + 1.11e6T + 5.06e11T^{2} \)
53 \( 1 + 7.71e5T + 1.17e12T^{2} \)
61 \( 1 + 2.11e5T + 3.14e12T^{2} \)
67 \( 1 - 1.48e6T + 6.06e12T^{2} \)
71 \( 1 - 3.40e6T + 9.09e12T^{2} \)
73 \( 1 + 3.93e6T + 1.10e13T^{2} \)
79 \( 1 - 4.55e6T + 1.92e13T^{2} \)
83 \( 1 + 4.51e6T + 2.71e13T^{2} \)
89 \( 1 + 4.21e6T + 4.42e13T^{2} \)
97 \( 1 + 1.59e6T + 8.07e13T^{2} \)
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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

−11.02827692371449041083492926623, −10.05543564901976296765453265960, −9.299220794185227296547684762011, −6.81014425327092757402600729179, −6.44812366549847386944124823255, −5.50422956070156694544572951764, −4.60893395553417150406032143802, −2.97535620155047033052839010204, −2.24430451568000857971306655278, 0, 2.24430451568000857971306655278, 2.97535620155047033052839010204, 4.60893395553417150406032143802, 5.50422956070156694544572951764, 6.44812366549847386944124823255, 6.81014425327092757402600729179, 9.299220794185227296547684762011, 10.05543564901976296765453265960, 11.02827692371449041083492926623

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