Properties

Label 2-177-1.1-c3-0-19
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $10.4433$
Root an. cond. $3.23161$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.0253·2-s − 3·3-s − 7.99·4-s + 8.85·5-s − 0.0759·6-s + 13.8·7-s − 0.405·8-s + 9·9-s + 0.224·10-s − 33.5·11-s + 23.9·12-s − 15.9·13-s + 0.351·14-s − 26.5·15-s + 63.9·16-s − 14.1·17-s + 0.227·18-s − 111.·19-s − 70.8·20-s − 41.6·21-s − 0.850·22-s − 188.·23-s + 1.21·24-s − 46.5·25-s − 0.403·26-s − 27·27-s − 110.·28-s + ⋯
L(s)  = 1  + 0.00895·2-s − 0.577·3-s − 0.999·4-s + 0.792·5-s − 0.00516·6-s + 0.749·7-s − 0.0179·8-s + 0.333·9-s + 0.00709·10-s − 0.920·11-s + 0.577·12-s − 0.339·13-s + 0.00670·14-s − 0.457·15-s + 0.999·16-s − 0.202·17-s + 0.00298·18-s − 1.35·19-s − 0.791·20-s − 0.432·21-s − 0.00824·22-s − 1.71·23-s + 0.0103·24-s − 0.372·25-s − 0.00304·26-s − 0.192·27-s − 0.749·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(10.4433\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ -1)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
59 \( 1 - 59T \)
good2 \( 1 - 0.0253T + 8T^{2} \)
5 \( 1 - 8.85T + 125T^{2} \)
7 \( 1 - 13.8T + 343T^{2} \)
11 \( 1 + 33.5T + 1.33e3T^{2} \)
13 \( 1 + 15.9T + 2.19e3T^{2} \)
17 \( 1 + 14.1T + 4.91e3T^{2} \)
19 \( 1 + 111.T + 6.85e3T^{2} \)
23 \( 1 + 188.T + 1.21e4T^{2} \)
29 \( 1 + 56.6T + 2.43e4T^{2} \)
31 \( 1 + 18.4T + 2.97e4T^{2} \)
37 \( 1 - 53.8T + 5.06e4T^{2} \)
41 \( 1 + 68.8T + 6.89e4T^{2} \)
43 \( 1 + 344.T + 7.95e4T^{2} \)
47 \( 1 - 291.T + 1.03e5T^{2} \)
53 \( 1 - 636.T + 1.48e5T^{2} \)
61 \( 1 + 489.T + 2.26e5T^{2} \)
67 \( 1 - 91.2T + 3.00e5T^{2} \)
71 \( 1 - 179.T + 3.57e5T^{2} \)
73 \( 1 + 414.T + 3.89e5T^{2} \)
79 \( 1 + 889.T + 4.93e5T^{2} \)
83 \( 1 - 812.T + 5.71e5T^{2} \)
89 \( 1 + 623.T + 7.04e5T^{2} \)
97 \( 1 - 1.22e3T + 9.12e5T^{2} \)
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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

−11.80838481604954969183156046538, −10.50544302499193807901332203984, −9.936836182178608323172351401943, −8.682757238213593302717555100404, −7.73871437727288268444485329254, −6.08959625881058652956953784163, −5.21201927568492211947405715683, −4.19778645602240867787858310538, −1.99700688626211368875194327715, 0, 1.99700688626211368875194327715, 4.19778645602240867787858310538, 5.21201927568492211947405715683, 6.08959625881058652956953784163, 7.73871437727288268444485329254, 8.682757238213593302717555100404, 9.936836182178608323172351401943, 10.50544302499193807901332203984, 11.80838481604954969183156046538

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