Properties

Label 2-177-1.1-c5-0-26
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  − 0.501·2-s − 9·3-s − 31.7·4-s − 14.0·5-s + 4.51·6-s + 117.·7-s + 31.9·8-s + 81·9-s + 7.04·10-s + 53.5·11-s + 285.·12-s − 471.·13-s − 58.8·14-s + 126.·15-s + 9.99e2·16-s + 1.16e3·17-s − 40.6·18-s + 1.82e3·19-s + 446.·20-s − 1.05e3·21-s − 26.8·22-s + 254.·23-s − 287.·24-s − 2.92e3·25-s + 236.·26-s − 729·27-s − 3.72e3·28-s + ⋯
L(s)  = 1  − 0.0886·2-s − 0.577·3-s − 0.992·4-s − 0.251·5-s + 0.0512·6-s + 0.904·7-s + 0.176·8-s + 0.333·9-s + 0.0222·10-s + 0.133·11-s + 0.572·12-s − 0.774·13-s − 0.0801·14-s + 0.145·15-s + 0.976·16-s + 0.977·17-s − 0.0295·18-s + 1.16·19-s + 0.249·20-s − 0.522·21-s − 0.0118·22-s + 0.100·23-s − 0.102·24-s − 0.936·25-s + 0.0686·26-s − 0.192·27-s − 0.897·28-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 \( 1 + 0.501T + 32T^{2} \)
5 \( 1 + 14.0T + 3.12e3T^{2} \)
7 \( 1 - 117.T + 1.68e4T^{2} \)
11 \( 1 - 53.5T + 1.61e5T^{2} \)
13 \( 1 + 471.T + 3.71e5T^{2} \)
17 \( 1 - 1.16e3T + 1.41e6T^{2} \)
19 \( 1 - 1.82e3T + 2.47e6T^{2} \)
23 \( 1 - 254.T + 6.43e6T^{2} \)
29 \( 1 + 6.59e3T + 2.05e7T^{2} \)
31 \( 1 - 1.10e3T + 2.86e7T^{2} \)
37 \( 1 + 7.74e3T + 6.93e7T^{2} \)
41 \( 1 + 5.82e3T + 1.15e8T^{2} \)
43 \( 1 + 6.40e3T + 1.47e8T^{2} \)
47 \( 1 + 2.85e3T + 2.29e8T^{2} \)
53 \( 1 + 1.66e4T + 4.18e8T^{2} \)
61 \( 1 - 3.70e4T + 8.44e8T^{2} \)
67 \( 1 - 1.09e4T + 1.35e9T^{2} \)
71 \( 1 + 1.54e4T + 1.80e9T^{2} \)
73 \( 1 + 4.21e4T + 2.07e9T^{2} \)
79 \( 1 + 8.96e4T + 3.07e9T^{2} \)
83 \( 1 - 5.86e4T + 3.93e9T^{2} \)
89 \( 1 - 6.17e4T + 5.58e9T^{2} \)
97 \( 1 + 1.38e4T + 8.58e9T^{2} \)
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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

−11.47250057806313265548177668321, −10.18695290815484730145457425979, −9.413756950308186690040968181818, −8.118700587364579361914324142701, −7.35576666733020258045283293496, −5.56992484267077749937967424986, −4.88636311218501977113863508411, −3.62407186945185970731829578793, −1.42824967717719035529577203089, 0, 1.42824967717719035529577203089, 3.62407186945185970731829578793, 4.88636311218501977113863508411, 5.56992484267077749937967424986, 7.35576666733020258045283293496, 8.118700587364579361914324142701, 9.413756950308186690040968181818, 10.18695290815484730145457425979, 11.47250057806313265548177668321

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