Properties

Label 2-177-1.1-c3-0-2
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.19·2-s − 3·3-s + 19.0·4-s − 8.30·5-s + 15.5·6-s + 21.5·7-s − 57.1·8-s + 9·9-s + 43.1·10-s − 28.3·11-s − 57.0·12-s − 28.2·13-s − 111.·14-s + 24.9·15-s + 145.·16-s + 21.8·17-s − 46.7·18-s − 122.·19-s − 157.·20-s − 64.6·21-s + 147.·22-s + 82.1·23-s + 171.·24-s − 56.0·25-s + 146.·26-s − 27·27-s + 409.·28-s + ⋯
L(s)  = 1  − 1.83·2-s − 0.577·3-s + 2.37·4-s − 0.742·5-s + 1.06·6-s + 1.16·7-s − 2.52·8-s + 0.333·9-s + 1.36·10-s − 0.777·11-s − 1.37·12-s − 0.601·13-s − 2.13·14-s + 0.428·15-s + 2.26·16-s + 0.311·17-s − 0.612·18-s − 1.47·19-s − 1.76·20-s − 0.671·21-s + 1.42·22-s + 0.744·23-s + 1.45·24-s − 0.448·25-s + 1.10·26-s − 0.192·27-s + 2.76·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4615016593\)
\(L(\frac12)\) \(\approx\) \(0.4615016593\)
\(L(\frac{5}{2})\) 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 \( 1 + 5.19T + 8T^{2} \)
5 \( 1 + 8.30T + 125T^{2} \)
7 \( 1 - 21.5T + 343T^{2} \)
11 \( 1 + 28.3T + 1.33e3T^{2} \)
13 \( 1 + 28.2T + 2.19e3T^{2} \)
17 \( 1 - 21.8T + 4.91e3T^{2} \)
19 \( 1 + 122.T + 6.85e3T^{2} \)
23 \( 1 - 82.1T + 1.21e4T^{2} \)
29 \( 1 - 86.9T + 2.43e4T^{2} \)
31 \( 1 - 131.T + 2.97e4T^{2} \)
37 \( 1 - 280.T + 5.06e4T^{2} \)
41 \( 1 - 381.T + 6.89e4T^{2} \)
43 \( 1 - 452.T + 7.95e4T^{2} \)
47 \( 1 + 158.T + 1.03e5T^{2} \)
53 \( 1 + 162.T + 1.48e5T^{2} \)
61 \( 1 + 368.T + 2.26e5T^{2} \)
67 \( 1 - 177.T + 3.00e5T^{2} \)
71 \( 1 - 58.9T + 3.57e5T^{2} \)
73 \( 1 - 880.T + 3.89e5T^{2} \)
79 \( 1 + 825.T + 4.93e5T^{2} \)
83 \( 1 - 1.42e3T + 5.71e5T^{2} \)
89 \( 1 - 1.55e3T + 7.04e5T^{2} \)
97 \( 1 - 1.81e3T + 9.12e5T^{2} \)
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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

−11.69758965620666144053040857389, −11.00082932430058605846862264539, −10.36847204527730390538517454634, −9.136720095602382159488128778291, −7.943345921992740838584923335550, −7.66358134356174094262280935708, −6.27005569144307146823893493993, −4.65582734937962653102870568504, −2.32616077856440584250832541071, −0.70692408628685967928988057097, 0.70692408628685967928988057097, 2.32616077856440584250832541071, 4.65582734937962653102870568504, 6.27005569144307146823893493993, 7.66358134356174094262280935708, 7.943345921992740838584923335550, 9.136720095602382159488128778291, 10.36847204527730390538517454634, 11.00082932430058605846862264539, 11.69758965620666144053040857389

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