Properties

Label 2-177-1.1-c7-0-7
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  + 3.56·2-s − 27·3-s − 115.·4-s − 210.·5-s − 96.3·6-s + 1.54e3·7-s − 868.·8-s + 729·9-s − 750.·10-s − 1.82e3·11-s + 3.11e3·12-s − 1.23e4·13-s + 5.51e3·14-s + 5.67e3·15-s + 1.16e4·16-s − 3.60e4·17-s + 2.60e3·18-s + 1.82e4·19-s + 2.42e4·20-s − 4.16e4·21-s − 6.49e3·22-s − 9.97e4·23-s + 2.34e4·24-s − 3.39e4·25-s − 4.41e4·26-s − 1.96e4·27-s − 1.78e5·28-s + ⋯
L(s)  = 1  + 0.315·2-s − 0.577·3-s − 0.900·4-s − 0.752·5-s − 0.182·6-s + 1.70·7-s − 0.599·8-s + 0.333·9-s − 0.237·10-s − 0.412·11-s + 0.519·12-s − 1.56·13-s + 0.536·14-s + 0.434·15-s + 0.711·16-s − 1.77·17-s + 0.105·18-s + 0.610·19-s + 0.677·20-s − 0.982·21-s − 0.130·22-s − 1.71·23-s + 0.346·24-s − 0.434·25-s − 0.492·26-s − 0.192·27-s − 1.53·28-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

\(L(4)\) \(\approx\) \(0.8775409307\)
\(L(\frac12)\) \(\approx\) \(0.8775409307\)
\(L(\frac{9}{2})\) 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 \( 1 - 3.56T + 128T^{2} \)
5 \( 1 + 210.T + 7.81e4T^{2} \)
7 \( 1 - 1.54e3T + 8.23e5T^{2} \)
11 \( 1 + 1.82e3T + 1.94e7T^{2} \)
13 \( 1 + 1.23e4T + 6.27e7T^{2} \)
17 \( 1 + 3.60e4T + 4.10e8T^{2} \)
19 \( 1 - 1.82e4T + 8.93e8T^{2} \)
23 \( 1 + 9.97e4T + 3.40e9T^{2} \)
29 \( 1 - 1.41e4T + 1.72e10T^{2} \)
31 \( 1 - 5.52e4T + 2.75e10T^{2} \)
37 \( 1 - 5.32e5T + 9.49e10T^{2} \)
41 \( 1 + 2.78e4T + 1.94e11T^{2} \)
43 \( 1 - 7.84e5T + 2.71e11T^{2} \)
47 \( 1 - 1.15e5T + 5.06e11T^{2} \)
53 \( 1 - 5.66e5T + 1.17e12T^{2} \)
61 \( 1 + 1.26e5T + 3.14e12T^{2} \)
67 \( 1 + 4.67e6T + 6.06e12T^{2} \)
71 \( 1 - 1.27e6T + 9.09e12T^{2} \)
73 \( 1 - 5.52e4T + 1.10e13T^{2} \)
79 \( 1 + 1.32e6T + 1.92e13T^{2} \)
83 \( 1 - 2.54e6T + 2.71e13T^{2} \)
89 \( 1 - 8.04e6T + 4.42e13T^{2} \)
97 \( 1 - 1.17e7T + 8.07e13T^{2} \)
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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

−11.65642148763858876955277657125, −10.57392392364465662835930399593, −9.372471339496776648195553919854, −8.120197036000921813583751925477, −7.52011500852097348649260517601, −5.76731091937095381704244399834, −4.61138129732780102837082068112, −4.33594940297041218513585019183, −2.23569727826342286127491536366, −0.49463714486490465142525450717, 0.49463714486490465142525450717, 2.23569727826342286127491536366, 4.33594940297041218513585019183, 4.61138129732780102837082068112, 5.76731091937095381704244399834, 7.52011500852097348649260517601, 8.120197036000921813583751925477, 9.372471339496776648195553919854, 10.57392392364465662835930399593, 11.65642148763858876955277657125

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