Properties

Label 2-177-1.1-c7-0-34
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  + 16.5·2-s − 27·3-s + 147.·4-s + 174.·5-s − 447.·6-s + 1.11e3·7-s + 319.·8-s + 729·9-s + 2.90e3·10-s + 3.45e3·11-s − 3.97e3·12-s − 368.·13-s + 1.84e4·14-s − 4.72e3·15-s − 1.35e4·16-s + 1.20e4·17-s + 1.20e4·18-s − 2.03e4·19-s + 2.57e4·20-s − 2.99e4·21-s + 5.73e4·22-s + 2.76e4·23-s − 8.62e3·24-s − 4.75e4·25-s − 6.11e3·26-s − 1.96e4·27-s + 1.63e5·28-s + ⋯
L(s)  = 1  + 1.46·2-s − 0.577·3-s + 1.15·4-s + 0.626·5-s − 0.846·6-s + 1.22·7-s + 0.220·8-s + 0.333·9-s + 0.918·10-s + 0.783·11-s − 0.664·12-s − 0.0465·13-s + 1.79·14-s − 0.361·15-s − 0.827·16-s + 0.593·17-s + 0.488·18-s − 0.682·19-s + 0.720·20-s − 0.706·21-s + 1.14·22-s + 0.474·23-s − 0.127·24-s − 0.608·25-s − 0.0682·26-s − 0.192·27-s + 1.40·28-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

\(L(4)\) \(\approx\) \(5.377793233\)
\(L(\frac12)\) \(\approx\) \(5.377793233\)
\(L(\frac{9}{2})\) 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 \( 1 - 16.5T + 128T^{2} \)
5 \( 1 - 174.T + 7.81e4T^{2} \)
7 \( 1 - 1.11e3T + 8.23e5T^{2} \)
11 \( 1 - 3.45e3T + 1.94e7T^{2} \)
13 \( 1 + 368.T + 6.27e7T^{2} \)
17 \( 1 - 1.20e4T + 4.10e8T^{2} \)
19 \( 1 + 2.03e4T + 8.93e8T^{2} \)
23 \( 1 - 2.76e4T + 3.40e9T^{2} \)
29 \( 1 - 1.41e5T + 1.72e10T^{2} \)
31 \( 1 - 2.24e5T + 2.75e10T^{2} \)
37 \( 1 - 4.33e5T + 9.49e10T^{2} \)
41 \( 1 - 2.86e5T + 1.94e11T^{2} \)
43 \( 1 + 1.52e5T + 2.71e11T^{2} \)
47 \( 1 - 4.49e5T + 5.06e11T^{2} \)
53 \( 1 - 1.41e5T + 1.17e12T^{2} \)
61 \( 1 + 5.22e5T + 3.14e12T^{2} \)
67 \( 1 - 1.89e6T + 6.06e12T^{2} \)
71 \( 1 - 3.16e6T + 9.09e12T^{2} \)
73 \( 1 - 3.37e6T + 1.10e13T^{2} \)
79 \( 1 + 1.64e6T + 1.92e13T^{2} \)
83 \( 1 + 6.36e6T + 2.71e13T^{2} \)
89 \( 1 - 5.43e6T + 4.42e13T^{2} \)
97 \( 1 + 2.91e6T + 8.07e13T^{2} \)
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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

−11.68402711840782443280612679514, −10.83992658316770169780812356929, −9.556279385537436520401952043710, −8.157330379075469906843986201609, −6.67844369929411066913512285429, −5.85557290746436087734334969153, −4.89224545348564307056459316963, −4.10340674757939942966180505212, −2.47764219970074304637001463125, −1.16199140779909799718345720033, 1.16199140779909799718345720033, 2.47764219970074304637001463125, 4.10340674757939942966180505212, 4.89224545348564307056459316963, 5.85557290746436087734334969153, 6.67844369929411066913512285429, 8.157330379075469906843986201609, 9.556279385537436520401952043710, 10.83992658316770169780812356929, 11.68402711840782443280612679514

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