Properties

Label 2-177-1.1-c5-0-33
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10.5·2-s + 9·3-s + 79.7·4-s − 45.7·5-s + 95.1·6-s − 1.90·7-s + 504.·8-s + 81·9-s − 483.·10-s + 342.·11-s + 717.·12-s + 540.·13-s − 20.1·14-s − 411.·15-s + 2.78e3·16-s + 551.·17-s + 856.·18-s + 1.65e3·19-s − 3.64e3·20-s − 17.1·21-s + 3.62e3·22-s − 1.80e3·23-s + 4.54e3·24-s − 1.03e3·25-s + 5.71e3·26-s + 729·27-s − 152.·28-s + ⋯
L(s)  = 1  + 1.86·2-s + 0.577·3-s + 2.49·4-s − 0.818·5-s + 1.07·6-s − 0.0147·7-s + 2.78·8-s + 0.333·9-s − 1.52·10-s + 0.853·11-s + 1.43·12-s + 0.886·13-s − 0.0274·14-s − 0.472·15-s + 2.71·16-s + 0.462·17-s + 0.622·18-s + 1.05·19-s − 2.04·20-s − 0.00849·21-s + 1.59·22-s − 0.711·23-s + 1.60·24-s − 0.329·25-s + 1.65·26-s + 0.192·27-s − 0.0366·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(7.335458503\)
\(L(\frac12)\) \(\approx\) \(7.335458503\)
\(L(\frac{7}{2})\) 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 \( 1 - 10.5T + 32T^{2} \)
5 \( 1 + 45.7T + 3.12e3T^{2} \)
7 \( 1 + 1.90T + 1.68e4T^{2} \)
11 \( 1 - 342.T + 1.61e5T^{2} \)
13 \( 1 - 540.T + 3.71e5T^{2} \)
17 \( 1 - 551.T + 1.41e6T^{2} \)
19 \( 1 - 1.65e3T + 2.47e6T^{2} \)
23 \( 1 + 1.80e3T + 6.43e6T^{2} \)
29 \( 1 - 2.64e3T + 2.05e7T^{2} \)
31 \( 1 + 2.43e3T + 2.86e7T^{2} \)
37 \( 1 + 1.16e4T + 6.93e7T^{2} \)
41 \( 1 + 1.36e4T + 1.15e8T^{2} \)
43 \( 1 + 1.36e4T + 1.47e8T^{2} \)
47 \( 1 + 5.33e3T + 2.29e8T^{2} \)
53 \( 1 - 9.60e3T + 4.18e8T^{2} \)
61 \( 1 - 4.25e3T + 8.44e8T^{2} \)
67 \( 1 - 1.24e4T + 1.35e9T^{2} \)
71 \( 1 - 5.73e4T + 1.80e9T^{2} \)
73 \( 1 - 2.18e4T + 2.07e9T^{2} \)
79 \( 1 + 5.71e4T + 3.07e9T^{2} \)
83 \( 1 + 8.05e4T + 3.93e9T^{2} \)
89 \( 1 + 6.03e4T + 5.58e9T^{2} \)
97 \( 1 - 1.52e5T + 8.58e9T^{2} \)
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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

−11.88727020364299404446143673112, −11.47053746623531067861423641749, −10.08886901149423696658612109021, −8.430828991830959526745945760115, −7.32544009685646385852340397698, −6.34715363075082078724856392146, −5.07235602929809305571324591756, −3.80291692756019742076441320280, −3.34464174502548026475166253483, −1.62590268383434929200191830188, 1.62590268383434929200191830188, 3.34464174502548026475166253483, 3.80291692756019742076441320280, 5.07235602929809305571324591756, 6.34715363075082078724856392146, 7.32544009685646385852340397698, 8.430828991830959526745945760115, 10.08886901149423696658612109021, 11.47053746623531067861423641749, 11.88727020364299404446143673112

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