Properties

Label 2-177-1.1-c7-0-60
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  + 17.9·2-s − 27·3-s + 193.·4-s + 1.22·5-s − 484.·6-s − 719.·7-s + 1.18e3·8-s + 729·9-s + 21.9·10-s + 1.11e3·11-s − 5.23e3·12-s + 8.28e3·13-s − 1.29e4·14-s − 33.0·15-s − 3.58e3·16-s + 4.83e3·17-s + 1.30e4·18-s − 3.16e4·19-s + 237.·20-s + 1.94e4·21-s + 1.99e4·22-s − 4.35e4·23-s − 3.19e4·24-s − 7.81e4·25-s + 1.48e5·26-s − 1.96e4·27-s − 1.39e5·28-s + ⋯
L(s)  = 1  + 1.58·2-s − 0.577·3-s + 1.51·4-s + 0.00437·5-s − 0.915·6-s − 0.792·7-s + 0.817·8-s + 0.333·9-s + 0.00694·10-s + 0.251·11-s − 0.875·12-s + 1.04·13-s − 1.25·14-s − 0.00252·15-s − 0.218·16-s + 0.238·17-s + 0.528·18-s − 1.05·19-s + 0.00663·20-s + 0.457·21-s + 0.399·22-s − 0.746·23-s − 0.472·24-s − 0.999·25-s + 1.65·26-s − 0.192·27-s − 1.20·28-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 \( 1 - 17.9T + 128T^{2} \)
5 \( 1 - 1.22T + 7.81e4T^{2} \)
7 \( 1 + 719.T + 8.23e5T^{2} \)
11 \( 1 - 1.11e3T + 1.94e7T^{2} \)
13 \( 1 - 8.28e3T + 6.27e7T^{2} \)
17 \( 1 - 4.83e3T + 4.10e8T^{2} \)
19 \( 1 + 3.16e4T + 8.93e8T^{2} \)
23 \( 1 + 4.35e4T + 3.40e9T^{2} \)
29 \( 1 + 2.32e5T + 1.72e10T^{2} \)
31 \( 1 + 8.53e4T + 2.75e10T^{2} \)
37 \( 1 + 5.27e4T + 9.49e10T^{2} \)
41 \( 1 + 5.89e5T + 1.94e11T^{2} \)
43 \( 1 - 3.67e5T + 2.71e11T^{2} \)
47 \( 1 - 7.39e5T + 5.06e11T^{2} \)
53 \( 1 - 1.53e6T + 1.17e12T^{2} \)
61 \( 1 - 3.25e5T + 3.14e12T^{2} \)
67 \( 1 + 1.72e6T + 6.06e12T^{2} \)
71 \( 1 + 4.29e6T + 9.09e12T^{2} \)
73 \( 1 + 1.55e6T + 1.10e13T^{2} \)
79 \( 1 + 3.39e6T + 1.92e13T^{2} \)
83 \( 1 + 7.04e6T + 2.71e13T^{2} \)
89 \( 1 - 1.14e7T + 4.42e13T^{2} \)
97 \( 1 - 1.36e7T + 8.07e13T^{2} \)
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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

−11.29681396378469317044961330783, −10.25343450965525419423317687242, −8.913921767428343817418970609080, −7.24358436474499742477705010155, −6.14239498243104375484921474110, −5.68205034291030480828999791834, −4.19435473682409357227715477557, −3.50933778246288196030918229703, −1.92961638747251886527349150656, 0, 1.92961638747251886527349150656, 3.50933778246288196030918229703, 4.19435473682409357227715477557, 5.68205034291030480828999791834, 6.14239498243104375484921474110, 7.24358436474499742477705010155, 8.913921767428343817418970609080, 10.25343450965525419423317687242, 11.29681396378469317044961330783

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