Properties

Label 2-177-1.1-c9-0-55
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $91.1613$
Root an. cond. $9.54784$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 31.8·2-s + 81·3-s + 504.·4-s − 1.88e3·5-s − 2.58e3·6-s + 1.22e4·7-s + 225.·8-s + 6.56e3·9-s + 6.02e4·10-s − 1.70e4·11-s + 4.08e4·12-s + 2.77e3·13-s − 3.91e5·14-s − 1.53e5·15-s − 2.65e5·16-s − 3.07e5·17-s − 2.09e5·18-s + 3.34e5·19-s − 9.53e5·20-s + 9.94e5·21-s + 5.43e5·22-s − 8.29e5·23-s + 1.82e4·24-s + 1.61e6·25-s − 8.83e4·26-s + 5.31e5·27-s + 6.19e6·28-s + ⋯
L(s)  = 1  − 1.40·2-s + 0.577·3-s + 0.986·4-s − 1.35·5-s − 0.813·6-s + 1.93·7-s + 0.0194·8-s + 0.333·9-s + 1.90·10-s − 0.351·11-s + 0.569·12-s + 0.0269·13-s − 2.72·14-s − 0.780·15-s − 1.01·16-s − 0.892·17-s − 0.469·18-s + 0.589·19-s − 1.33·20-s + 1.11·21-s + 0.494·22-s − 0.618·23-s + 0.0112·24-s + 0.827·25-s − 0.0379·26-s + 0.192·27-s + 1.90·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(91.1613\)
Root analytic conductor: \(9.54784\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 81T \)
59 \( 1 - 1.21e7T \)
good2 \( 1 + 31.8T + 512T^{2} \)
5 \( 1 + 1.88e3T + 1.95e6T^{2} \)
7 \( 1 - 1.22e4T + 4.03e7T^{2} \)
11 \( 1 + 1.70e4T + 2.35e9T^{2} \)
13 \( 1 - 2.77e3T + 1.06e10T^{2} \)
17 \( 1 + 3.07e5T + 1.18e11T^{2} \)
19 \( 1 - 3.34e5T + 3.22e11T^{2} \)
23 \( 1 + 8.29e5T + 1.80e12T^{2} \)
29 \( 1 + 1.06e6T + 1.45e13T^{2} \)
31 \( 1 + 8.06e5T + 2.64e13T^{2} \)
37 \( 1 + 2.98e5T + 1.29e14T^{2} \)
41 \( 1 - 9.57e5T + 3.27e14T^{2} \)
43 \( 1 + 1.34e7T + 5.02e14T^{2} \)
47 \( 1 - 2.47e6T + 1.11e15T^{2} \)
53 \( 1 + 1.59e7T + 3.29e15T^{2} \)
61 \( 1 - 4.27e7T + 1.16e16T^{2} \)
67 \( 1 + 1.05e8T + 2.72e16T^{2} \)
71 \( 1 + 2.00e6T + 4.58e16T^{2} \)
73 \( 1 - 4.31e8T + 5.88e16T^{2} \)
79 \( 1 + 3.46e8T + 1.19e17T^{2} \)
83 \( 1 - 2.37e8T + 1.86e17T^{2} \)
89 \( 1 - 5.27e8T + 3.50e17T^{2} \)
97 \( 1 - 9.13e8T + 7.60e17T^{2} \)
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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

−10.55477076684853104955072928587, −9.192806650460643580872783284709, −8.228652722409017783625495863881, −7.953673999707880210361994886026, −7.13206375873013511654360777662, −4.92703636913103047486797399963, −3.99796734962131603356522808092, −2.23817075882151003970999567278, −1.21602035915372520097497812932, 0, 1.21602035915372520097497812932, 2.23817075882151003970999567278, 3.99796734962131603356522808092, 4.92703636913103047486797399963, 7.13206375873013511654360777662, 7.953673999707880210361994886026, 8.228652722409017783625495863881, 9.192806650460643580872783284709, 10.55477076684853104955072928587

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