Properties

Label 2-177-1.1-c5-0-30
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  − 10.9·2-s − 9·3-s + 86.8·4-s + 88.1·5-s + 98.1·6-s + 61.7·7-s − 598.·8-s + 81·9-s − 960.·10-s + 423.·11-s − 781.·12-s − 1.04e3·13-s − 672.·14-s − 793.·15-s + 3.74e3·16-s − 2.03e3·17-s − 883.·18-s − 1.53e3·19-s + 7.65e3·20-s − 555.·21-s − 4.61e3·22-s + 2.98e3·23-s + 5.38e3·24-s + 4.63e3·25-s + 1.14e4·26-s − 729·27-s + 5.36e3·28-s + ⋯
L(s)  = 1  − 1.92·2-s − 0.577·3-s + 2.71·4-s + 1.57·5-s + 1.11·6-s + 0.476·7-s − 3.30·8-s + 0.333·9-s − 3.03·10-s + 1.05·11-s − 1.56·12-s − 1.71·13-s − 0.917·14-s − 0.910·15-s + 3.65·16-s − 1.70·17-s − 0.642·18-s − 0.973·19-s + 4.27·20-s − 0.274·21-s − 2.03·22-s + 1.17·23-s + 1.90·24-s + 1.48·25-s + 3.31·26-s − 0.192·27-s + 1.29·28-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 \( 1 + 10.9T + 32T^{2} \)
5 \( 1 - 88.1T + 3.12e3T^{2} \)
7 \( 1 - 61.7T + 1.68e4T^{2} \)
11 \( 1 - 423.T + 1.61e5T^{2} \)
13 \( 1 + 1.04e3T + 3.71e5T^{2} \)
17 \( 1 + 2.03e3T + 1.41e6T^{2} \)
19 \( 1 + 1.53e3T + 2.47e6T^{2} \)
23 \( 1 - 2.98e3T + 6.43e6T^{2} \)
29 \( 1 - 115.T + 2.05e7T^{2} \)
31 \( 1 + 3.62e3T + 2.86e7T^{2} \)
37 \( 1 + 1.50e4T + 6.93e7T^{2} \)
41 \( 1 + 829.T + 1.15e8T^{2} \)
43 \( 1 + 1.22e4T + 1.47e8T^{2} \)
47 \( 1 + 1.61e4T + 2.29e8T^{2} \)
53 \( 1 - 3.44e4T + 4.18e8T^{2} \)
61 \( 1 - 1.64e4T + 8.44e8T^{2} \)
67 \( 1 - 9.81e3T + 1.35e9T^{2} \)
71 \( 1 - 3.38e3T + 1.80e9T^{2} \)
73 \( 1 - 2.70e4T + 2.07e9T^{2} \)
79 \( 1 - 2.04e4T + 3.07e9T^{2} \)
83 \( 1 + 5.45e4T + 3.93e9T^{2} \)
89 \( 1 + 6.97e4T + 5.58e9T^{2} \)
97 \( 1 - 4.85e4T + 8.58e9T^{2} \)
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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

−10.91195880880638164485399721657, −10.12218538800775077347514535740, −9.321356182542672203668874221968, −8.648770108381383298580749338193, −6.94645224988009614124550588068, −6.59756799380572094058236864619, −5.15690135088567038919501953935, −2.30381454022528759045812443170, −1.58418288688224332054055992137, 0, 1.58418288688224332054055992137, 2.30381454022528759045812443170, 5.15690135088567038919501953935, 6.59756799380572094058236864619, 6.94645224988009614124550588068, 8.648770108381383298580749338193, 9.321356182542672203668874221968, 10.12218538800775077347514535740, 10.91195880880638164485399721657

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