Properties

Label 2-177-1.1-c7-0-45
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  + 2.11·2-s + 27·3-s − 123.·4-s − 537.·5-s + 57.0·6-s + 1.25e3·7-s − 531.·8-s + 729·9-s − 1.13e3·10-s + 7.73e3·11-s − 3.33e3·12-s − 8.86e3·13-s + 2.66e3·14-s − 1.45e4·15-s + 1.46e4·16-s + 2.23e4·17-s + 1.54e3·18-s − 2.48e4·19-s + 6.63e4·20-s + 3.39e4·21-s + 1.63e4·22-s − 2.66e4·23-s − 1.43e4·24-s + 2.10e5·25-s − 1.87e4·26-s + 1.96e4·27-s − 1.55e5·28-s + ⋯
L(s)  = 1  + 0.186·2-s + 0.577·3-s − 0.965·4-s − 1.92·5-s + 0.107·6-s + 1.38·7-s − 0.367·8-s + 0.333·9-s − 0.358·10-s + 1.75·11-s − 0.557·12-s − 1.11·13-s + 0.259·14-s − 1.10·15-s + 0.896·16-s + 1.10·17-s + 0.0622·18-s − 0.832·19-s + 1.85·20-s + 0.801·21-s + 0.327·22-s − 0.457·23-s − 0.211·24-s + 2.69·25-s − 0.208·26-s + 0.192·27-s − 1.33·28-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 \( 1 - 2.11T + 128T^{2} \)
5 \( 1 + 537.T + 7.81e4T^{2} \)
7 \( 1 - 1.25e3T + 8.23e5T^{2} \)
11 \( 1 - 7.73e3T + 1.94e7T^{2} \)
13 \( 1 + 8.86e3T + 6.27e7T^{2} \)
17 \( 1 - 2.23e4T + 4.10e8T^{2} \)
19 \( 1 + 2.48e4T + 8.93e8T^{2} \)
23 \( 1 + 2.66e4T + 3.40e9T^{2} \)
29 \( 1 - 1.01e4T + 1.72e10T^{2} \)
31 \( 1 + 2.65e5T + 2.75e10T^{2} \)
37 \( 1 - 1.22e5T + 9.49e10T^{2} \)
41 \( 1 + 7.41e5T + 1.94e11T^{2} \)
43 \( 1 + 1.66e5T + 2.71e11T^{2} \)
47 \( 1 + 5.45e5T + 5.06e11T^{2} \)
53 \( 1 + 1.54e6T + 1.17e12T^{2} \)
61 \( 1 - 2.74e6T + 3.14e12T^{2} \)
67 \( 1 + 4.16e4T + 6.06e12T^{2} \)
71 \( 1 + 1.57e6T + 9.09e12T^{2} \)
73 \( 1 - 5.59e6T + 1.10e13T^{2} \)
79 \( 1 + 3.74e6T + 1.92e13T^{2} \)
83 \( 1 + 3.37e6T + 2.71e13T^{2} \)
89 \( 1 + 5.43e6T + 4.42e13T^{2} \)
97 \( 1 + 1.14e7T + 8.07e13T^{2} \)
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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

−11.18892171514544611569098543994, −9.676690688878455271375102244853, −8.532043606659255280215385745977, −8.064297417533507565978359285125, −7.08175108928917212567321206627, −4.99640270247791462484472408102, −4.18393982079452749094897476984, −3.50300788521764384675877225633, −1.39381011471324546701315669020, 0, 1.39381011471324546701315669020, 3.50300788521764384675877225633, 4.18393982079452749094897476984, 4.99640270247791462484472408102, 7.08175108928917212567321206627, 8.064297417533507565978359285125, 8.532043606659255280215385745977, 9.676690688878455271375102244853, 11.18892171514544611569098543994

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