Properties

Label 2-177-1.1-c7-0-3
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.14·2-s − 27·3-s − 44.4·4-s − 27.9·5-s − 246.·6-s − 1.41e3·7-s − 1.57e3·8-s + 729·9-s − 255.·10-s − 4.88e3·11-s + 1.20e3·12-s − 5.21e3·13-s − 1.29e4·14-s + 755.·15-s − 8.71e3·16-s + 5.79e3·17-s + 6.66e3·18-s + 2.49e4·19-s + 1.24e3·20-s + 3.82e4·21-s − 4.46e4·22-s − 6.30e4·23-s + 4.25e4·24-s − 7.73e4·25-s − 4.76e4·26-s − 1.96e4·27-s + 6.29e4·28-s + ⋯
L(s)  = 1  + 0.807·2-s − 0.577·3-s − 0.347·4-s − 0.100·5-s − 0.466·6-s − 1.56·7-s − 1.08·8-s + 0.333·9-s − 0.0808·10-s − 1.10·11-s + 0.200·12-s − 0.657·13-s − 1.26·14-s + 0.0577·15-s − 0.532·16-s + 0.286·17-s + 0.269·18-s + 0.834·19-s + 0.0347·20-s + 0.900·21-s − 0.894·22-s − 1.07·23-s + 0.628·24-s − 0.989·25-s − 0.531·26-s − 0.192·27-s + 0.541·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(0.6210600909\)
\(L(\frac12)\) \(\approx\) \(0.6210600909\)
\(L(\frac{9}{2})\) 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 \( 1 - 9.14T + 128T^{2} \)
5 \( 1 + 27.9T + 7.81e4T^{2} \)
7 \( 1 + 1.41e3T + 8.23e5T^{2} \)
11 \( 1 + 4.88e3T + 1.94e7T^{2} \)
13 \( 1 + 5.21e3T + 6.27e7T^{2} \)
17 \( 1 - 5.79e3T + 4.10e8T^{2} \)
19 \( 1 - 2.49e4T + 8.93e8T^{2} \)
23 \( 1 + 6.30e4T + 3.40e9T^{2} \)
29 \( 1 - 8.65e4T + 1.72e10T^{2} \)
31 \( 1 - 2.68e5T + 2.75e10T^{2} \)
37 \( 1 + 2.35e5T + 9.49e10T^{2} \)
41 \( 1 + 5.85e5T + 1.94e11T^{2} \)
43 \( 1 + 8.06e5T + 2.71e11T^{2} \)
47 \( 1 - 5.57e5T + 5.06e11T^{2} \)
53 \( 1 - 2.07e6T + 1.17e12T^{2} \)
61 \( 1 + 1.17e6T + 3.14e12T^{2} \)
67 \( 1 - 4.57e6T + 6.06e12T^{2} \)
71 \( 1 - 7.98e5T + 9.09e12T^{2} \)
73 \( 1 + 2.35e6T + 1.10e13T^{2} \)
79 \( 1 - 1.17e5T + 1.92e13T^{2} \)
83 \( 1 - 2.44e6T + 2.71e13T^{2} \)
89 \( 1 + 5.83e6T + 4.42e13T^{2} \)
97 \( 1 + 6.38e6T + 8.07e13T^{2} \)
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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

−11.93854019282286311835674411455, −10.08269218348109756383264229432, −9.844297351029001363434653461255, −8.286308105664343549790358642634, −6.89558525658888418915818688372, −5.87345882802063846167874262341, −5.03676081702764973010995737216, −3.74645108603195601884051737046, −2.72473741866954551555730866720, −0.37132349566897025727966174151, 0.37132349566897025727966174151, 2.72473741866954551555730866720, 3.74645108603195601884051737046, 5.03676081702764973010995737216, 5.87345882802063846167874262341, 6.89558525658888418915818688372, 8.286308105664343549790358642634, 9.844297351029001363434653461255, 10.08269218348109756383264229432, 11.93854019282286311835674411455

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