Properties

Label 2-177-1.1-c3-0-12
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  − 3.06·2-s − 3·3-s + 1.36·4-s + 0.675·5-s + 9.18·6-s − 3.38·7-s + 20.3·8-s + 9·9-s − 2.06·10-s − 8.79·11-s − 4.09·12-s + 59.4·13-s + 10.3·14-s − 2.02·15-s − 73.0·16-s + 31.2·17-s − 27.5·18-s − 77.5·19-s + 0.921·20-s + 10.1·21-s + 26.9·22-s + 73.1·23-s − 60.9·24-s − 124.·25-s − 181.·26-s − 27·27-s − 4.62·28-s + ⋯
L(s)  = 1  − 1.08·2-s − 0.577·3-s + 0.170·4-s + 0.0603·5-s + 0.624·6-s − 0.182·7-s + 0.897·8-s + 0.333·9-s − 0.0653·10-s − 0.240·11-s − 0.0984·12-s + 1.26·13-s + 0.197·14-s − 0.0348·15-s − 1.14·16-s + 0.445·17-s − 0.360·18-s − 0.936·19-s + 0.0102·20-s + 0.105·21-s + 0.260·22-s + 0.662·23-s − 0.518·24-s − 0.996·25-s − 1.37·26-s − 0.192·27-s − 0.0311·28-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 \( 1 + 3.06T + 8T^{2} \)
5 \( 1 - 0.675T + 125T^{2} \)
7 \( 1 + 3.38T + 343T^{2} \)
11 \( 1 + 8.79T + 1.33e3T^{2} \)
13 \( 1 - 59.4T + 2.19e3T^{2} \)
17 \( 1 - 31.2T + 4.91e3T^{2} \)
19 \( 1 + 77.5T + 6.85e3T^{2} \)
23 \( 1 - 73.1T + 1.21e4T^{2} \)
29 \( 1 + 61.5T + 2.43e4T^{2} \)
31 \( 1 + 278.T + 2.97e4T^{2} \)
37 \( 1 + 326.T + 5.06e4T^{2} \)
41 \( 1 - 453.T + 6.89e4T^{2} \)
43 \( 1 + 174.T + 7.95e4T^{2} \)
47 \( 1 + 480.T + 1.03e5T^{2} \)
53 \( 1 + 185.T + 1.48e5T^{2} \)
61 \( 1 - 807.T + 2.26e5T^{2} \)
67 \( 1 + 129.T + 3.00e5T^{2} \)
71 \( 1 + 272.T + 3.57e5T^{2} \)
73 \( 1 + 610.T + 3.89e5T^{2} \)
79 \( 1 - 886.T + 4.93e5T^{2} \)
83 \( 1 - 51.4T + 5.71e5T^{2} \)
89 \( 1 - 182.T + 7.04e5T^{2} \)
97 \( 1 + 1.35e3T + 9.12e5T^{2} \)
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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

−11.29728290770596157949804437137, −10.69925077714069041598034539928, −9.699291551564355372952621454759, −8.763962774311562882736863978968, −7.78661866188301715337005269301, −6.60248344227984568182085980443, −5.35442008171096690081168049259, −3.85620392315917456670127358869, −1.57937202228929353460327924591, 0, 1.57937202228929353460327924591, 3.85620392315917456670127358869, 5.35442008171096690081168049259, 6.60248344227984568182085980443, 7.78661866188301715337005269301, 8.763962774311562882736863978968, 9.699291551564355372952621454759, 10.69925077714069041598034539928, 11.29728290770596157949804437137

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