Properties

Label 2-177-1.1-c5-0-24
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $28.3879$
Root an. cond. $5.32803$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.2·2-s + 9·3-s + 72.4·4-s − 99.2·5-s − 91.9·6-s + 109.·7-s − 413.·8-s + 81·9-s + 1.01e3·10-s − 193.·11-s + 651.·12-s + 577.·13-s − 1.12e3·14-s − 893.·15-s + 1.90e3·16-s − 332.·17-s − 827.·18-s − 2.11e3·19-s − 7.18e3·20-s + 989.·21-s + 1.98e3·22-s + 3.18e3·23-s − 3.71e3·24-s + 6.72e3·25-s − 5.90e3·26-s + 729·27-s + 7.96e3·28-s + ⋯
L(s)  = 1  − 1.80·2-s + 0.577·3-s + 2.26·4-s − 1.77·5-s − 1.04·6-s + 0.848·7-s − 2.28·8-s + 0.333·9-s + 3.20·10-s − 0.482·11-s + 1.30·12-s + 0.947·13-s − 1.53·14-s − 1.02·15-s + 1.86·16-s − 0.278·17-s − 0.602·18-s − 1.34·19-s − 4.01·20-s + 0.489·21-s + 0.872·22-s + 1.25·23-s − 1.31·24-s + 2.15·25-s − 1.71·26-s + 0.192·27-s + 1.92·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(28.3879\)
Root analytic conductor: \(5.32803\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 9T \)
59 \( 1 - 3.48e3T \)
good2 \( 1 + 10.2T + 32T^{2} \)
5 \( 1 + 99.2T + 3.12e3T^{2} \)
7 \( 1 - 109.T + 1.68e4T^{2} \)
11 \( 1 + 193.T + 1.61e5T^{2} \)
13 \( 1 - 577.T + 3.71e5T^{2} \)
17 \( 1 + 332.T + 1.41e6T^{2} \)
19 \( 1 + 2.11e3T + 2.47e6T^{2} \)
23 \( 1 - 3.18e3T + 6.43e6T^{2} \)
29 \( 1 - 6.11e3T + 2.05e7T^{2} \)
31 \( 1 + 2.02e3T + 2.86e7T^{2} \)
37 \( 1 - 3.58e3T + 6.93e7T^{2} \)
41 \( 1 - 4.11e3T + 1.15e8T^{2} \)
43 \( 1 + 2.31e4T + 1.47e8T^{2} \)
47 \( 1 + 6.68e3T + 2.29e8T^{2} \)
53 \( 1 - 1.82e4T + 4.18e8T^{2} \)
61 \( 1 + 1.48e4T + 8.44e8T^{2} \)
67 \( 1 + 1.08e4T + 1.35e9T^{2} \)
71 \( 1 + 8.00e4T + 1.80e9T^{2} \)
73 \( 1 + 2.21e4T + 2.07e9T^{2} \)
79 \( 1 + 5.75e4T + 3.07e9T^{2} \)
83 \( 1 + 9.89e4T + 3.93e9T^{2} \)
89 \( 1 + 9.17e4T + 5.58e9T^{2} \)
97 \( 1 - 4.91e4T + 8.58e9T^{2} \)
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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

−11.09701141007024413235792114221, −10.35530115461172407904095910663, −8.607104166788973086084062886076, −8.550748221969892406030945225019, −7.63723952431339310338092019144, −6.74551023922936909780634946126, −4.43686460579737305277955505759, −2.93966799828149235409535210629, −1.32676163948764000618424987288, 0, 1.32676163948764000618424987288, 2.93966799828149235409535210629, 4.43686460579737305277955505759, 6.74551023922936909780634946126, 7.63723952431339310338092019144, 8.550748221969892406030945225019, 8.607104166788973086084062886076, 10.35530115461172407904095910663, 11.09701141007024413235792114221

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