Properties

Label 2-177-1.1-c7-0-63
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  + 15.5·2-s + 27·3-s + 113.·4-s − 141.·5-s + 419.·6-s + 105.·7-s − 232.·8-s + 729·9-s − 2.20e3·10-s − 555.·11-s + 3.05e3·12-s − 1.12e4·13-s + 1.63e3·14-s − 3.82e3·15-s − 1.80e4·16-s − 3.13e3·17-s + 1.13e4·18-s − 2.19e4·19-s − 1.60e4·20-s + 2.84e3·21-s − 8.61e3·22-s + 2.85e4·23-s − 6.27e3·24-s − 5.80e4·25-s − 1.75e5·26-s + 1.96e4·27-s + 1.19e4·28-s + ⋯
L(s)  = 1  + 1.37·2-s + 0.577·3-s + 0.883·4-s − 0.507·5-s + 0.792·6-s + 0.116·7-s − 0.160·8-s + 0.333·9-s − 0.696·10-s − 0.125·11-s + 0.509·12-s − 1.42·13-s + 0.159·14-s − 0.292·15-s − 1.10·16-s − 0.155·17-s + 0.457·18-s − 0.734·19-s − 0.448·20-s + 0.0669·21-s − 0.172·22-s + 0.489·23-s − 0.0925·24-s − 0.742·25-s − 1.95·26-s + 0.192·27-s + 0.102·28-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 \( 1 - 15.5T + 128T^{2} \)
5 \( 1 + 141.T + 7.81e4T^{2} \)
7 \( 1 - 105.T + 8.23e5T^{2} \)
11 \( 1 + 555.T + 1.94e7T^{2} \)
13 \( 1 + 1.12e4T + 6.27e7T^{2} \)
17 \( 1 + 3.13e3T + 4.10e8T^{2} \)
19 \( 1 + 2.19e4T + 8.93e8T^{2} \)
23 \( 1 - 2.85e4T + 3.40e9T^{2} \)
29 \( 1 - 1.60e5T + 1.72e10T^{2} \)
31 \( 1 + 1.63e5T + 2.75e10T^{2} \)
37 \( 1 + 1.31e5T + 9.49e10T^{2} \)
41 \( 1 + 2.77e5T + 1.94e11T^{2} \)
43 \( 1 - 6.21e5T + 2.71e11T^{2} \)
47 \( 1 + 1.39e6T + 5.06e11T^{2} \)
53 \( 1 - 1.49e6T + 1.17e12T^{2} \)
61 \( 1 + 2.41e6T + 3.14e12T^{2} \)
67 \( 1 - 2.34e6T + 6.06e12T^{2} \)
71 \( 1 - 5.27e4T + 9.09e12T^{2} \)
73 \( 1 - 1.12e6T + 1.10e13T^{2} \)
79 \( 1 - 3.33e6T + 1.92e13T^{2} \)
83 \( 1 + 5.69e6T + 2.71e13T^{2} \)
89 \( 1 + 5.02e6T + 4.42e13T^{2} \)
97 \( 1 - 9.04e6T + 8.07e13T^{2} \)
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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

−11.26288409037874652679445840601, −9.943388084527682713046138977871, −8.773913528934636205357592235177, −7.60381644123269292813917401866, −6.56529461809468707792859965700, −5.14621141561087267934693102837, −4.33076859468340250288943094346, −3.22829725447923637400353294027, −2.16478968546648252159081541546, 0, 2.16478968546648252159081541546, 3.22829725447923637400353294027, 4.33076859468340250288943094346, 5.14621141561087267934693102837, 6.56529461809468707792859965700, 7.60381644123269292813917401866, 8.773913528934636205357592235177, 9.943388084527682713046138977871, 11.26288409037874652679445840601

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