Properties

Label 2-177-1.1-c7-0-40
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  − 15.0·2-s + 27·3-s + 97.4·4-s + 6.55·5-s − 405.·6-s + 1.46e3·7-s + 458.·8-s + 729·9-s − 98.3·10-s + 4.59e3·11-s + 2.63e3·12-s + 1.52e4·13-s − 2.20e4·14-s + 176.·15-s − 1.93e4·16-s + 3.89e4·17-s − 1.09e4·18-s − 499.·19-s + 638.·20-s + 3.95e4·21-s − 6.90e4·22-s + 5.82e4·23-s + 1.23e4·24-s − 7.80e4·25-s − 2.28e5·26-s + 1.96e4·27-s + 1.42e5·28-s + ⋯
L(s)  = 1  − 1.32·2-s + 0.577·3-s + 0.761·4-s + 0.0234·5-s − 0.766·6-s + 1.61·7-s + 0.316·8-s + 0.333·9-s − 0.0311·10-s + 1.04·11-s + 0.439·12-s + 1.92·13-s − 2.14·14-s + 0.0135·15-s − 1.18·16-s + 1.92·17-s − 0.442·18-s − 0.0167·19-s + 0.0178·20-s + 0.932·21-s − 1.38·22-s + 0.998·23-s + 0.182·24-s − 0.999·25-s − 2.55·26-s + 0.192·27-s + 1.23·28-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

\(L(4)\) \(\approx\) \(2.154221295\)
\(L(\frac12)\) \(\approx\) \(2.154221295\)
\(L(\frac{9}{2})\) 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 \( 1 + 15.0T + 128T^{2} \)
5 \( 1 - 6.55T + 7.81e4T^{2} \)
7 \( 1 - 1.46e3T + 8.23e5T^{2} \)
11 \( 1 - 4.59e3T + 1.94e7T^{2} \)
13 \( 1 - 1.52e4T + 6.27e7T^{2} \)
17 \( 1 - 3.89e4T + 4.10e8T^{2} \)
19 \( 1 + 499.T + 8.93e8T^{2} \)
23 \( 1 - 5.82e4T + 3.40e9T^{2} \)
29 \( 1 - 2.11e5T + 1.72e10T^{2} \)
31 \( 1 - 6.18e4T + 2.75e10T^{2} \)
37 \( 1 + 6.29e4T + 9.49e10T^{2} \)
41 \( 1 + 6.19e5T + 1.94e11T^{2} \)
43 \( 1 + 3.56e5T + 2.71e11T^{2} \)
47 \( 1 + 9.53e5T + 5.06e11T^{2} \)
53 \( 1 + 1.97e6T + 1.17e12T^{2} \)
61 \( 1 + 2.98e6T + 3.14e12T^{2} \)
67 \( 1 + 1.04e6T + 6.06e12T^{2} \)
71 \( 1 - 4.10e6T + 9.09e12T^{2} \)
73 \( 1 + 2.60e5T + 1.10e13T^{2} \)
79 \( 1 + 1.33e6T + 1.92e13T^{2} \)
83 \( 1 - 8.30e6T + 2.71e13T^{2} \)
89 \( 1 - 3.49e6T + 4.42e13T^{2} \)
97 \( 1 + 1.00e7T + 8.07e13T^{2} \)
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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

−11.15543730707037320303446967532, −10.23114839830096505879882573011, −9.167085213614107021945864648784, −8.259760164147606593780773430040, −7.943920646519697852069834320750, −6.50428894468619726509360889424, −4.85461455332262024116488799247, −3.49482336591809486354092833607, −1.42180224023623939772885336295, −1.27412100385318772932881929018, 1.27412100385318772932881929018, 1.42180224023623939772885336295, 3.49482336591809486354092833607, 4.85461455332262024116488799247, 6.50428894468619726509360889424, 7.943920646519697852069834320750, 8.259760164147606593780773430040, 9.167085213614107021945864648784, 10.23114839830096505879882573011, 11.15543730707037320303446967532

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