Properties

Label 2-177-1.1-c5-0-29
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  + 7.43·2-s + 9·3-s + 23.2·4-s + 48.3·5-s + 66.8·6-s + 120.·7-s − 65.0·8-s + 81·9-s + 359.·10-s − 192.·11-s + 209.·12-s + 600.·13-s + 895.·14-s + 434.·15-s − 1.22e3·16-s + 2.19e3·17-s + 602.·18-s − 393.·19-s + 1.12e3·20-s + 1.08e3·21-s − 1.42e3·22-s − 212.·23-s − 585.·24-s − 791.·25-s + 4.46e3·26-s + 729·27-s + 2.80e3·28-s + ⋯
L(s)  = 1  + 1.31·2-s + 0.577·3-s + 0.726·4-s + 0.864·5-s + 0.758·6-s + 0.929·7-s − 0.359·8-s + 0.333·9-s + 1.13·10-s − 0.478·11-s + 0.419·12-s + 0.985·13-s + 1.22·14-s + 0.498·15-s − 1.19·16-s + 1.84·17-s + 0.437·18-s − 0.250·19-s + 0.627·20-s + 0.536·21-s − 0.628·22-s − 0.0837·23-s − 0.207·24-s − 0.253·25-s + 1.29·26-s + 0.192·27-s + 0.675·28-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

\(L(3)\) \(\approx\) \(5.892554525\)
\(L(\frac12)\) \(\approx\) \(5.892554525\)
\(L(\frac{7}{2})\) 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 \( 1 - 7.43T + 32T^{2} \)
5 \( 1 - 48.3T + 3.12e3T^{2} \)
7 \( 1 - 120.T + 1.68e4T^{2} \)
11 \( 1 + 192.T + 1.61e5T^{2} \)
13 \( 1 - 600.T + 3.71e5T^{2} \)
17 \( 1 - 2.19e3T + 1.41e6T^{2} \)
19 \( 1 + 393.T + 2.47e6T^{2} \)
23 \( 1 + 212.T + 6.43e6T^{2} \)
29 \( 1 + 320.T + 2.05e7T^{2} \)
31 \( 1 + 1.19e3T + 2.86e7T^{2} \)
37 \( 1 - 1.86e3T + 6.93e7T^{2} \)
41 \( 1 + 1.01e4T + 1.15e8T^{2} \)
43 \( 1 - 1.82e4T + 1.47e8T^{2} \)
47 \( 1 - 1.90e4T + 2.29e8T^{2} \)
53 \( 1 + 1.84e4T + 4.18e8T^{2} \)
61 \( 1 + 1.64e4T + 8.44e8T^{2} \)
67 \( 1 - 1.03e4T + 1.35e9T^{2} \)
71 \( 1 + 1.63e4T + 1.80e9T^{2} \)
73 \( 1 + 6.88e4T + 2.07e9T^{2} \)
79 \( 1 + 3.49e3T + 3.07e9T^{2} \)
83 \( 1 + 9.53e4T + 3.93e9T^{2} \)
89 \( 1 + 1.99e4T + 5.58e9T^{2} \)
97 \( 1 + 9.96e4T + 8.58e9T^{2} \)
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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

−12.13773253111712578924451293433, −10.98852980215542899110532778907, −9.852064008397132147266880205716, −8.692482695374319520999874274557, −7.60754834032143041625478830339, −6.02610282570134457233307473999, −5.32794351510926806847343511298, −4.07336975055436216838445241926, −2.88142861647599151177250001050, −1.53364351533271335449056059127, 1.53364351533271335449056059127, 2.88142861647599151177250001050, 4.07336975055436216838445241926, 5.32794351510926806847343511298, 6.02610282570134457233307473999, 7.60754834032143041625478830339, 8.692482695374319520999874274557, 9.852064008397132147266880205716, 10.98852980215542899110532778907, 12.13773253111712578924451293433

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