Properties

Label 2-177-1.1-c7-0-29
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $55.2921$
Root an. cond. $7.43586$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 16.9·2-s + 27·3-s + 158.·4-s − 421.·5-s + 457.·6-s − 62.1·7-s + 517.·8-s + 729·9-s − 7.12e3·10-s + 8.42e3·11-s + 4.28e3·12-s + 1.13e4·13-s − 1.05e3·14-s − 1.13e4·15-s − 1.15e4·16-s + 1.30e3·17-s + 1.23e4·18-s − 7.90e3·19-s − 6.67e4·20-s − 1.67e3·21-s + 1.42e5·22-s − 2.03e3·23-s + 1.39e4·24-s + 9.91e4·25-s + 1.92e5·26-s + 1.96e4·27-s − 9.85e3·28-s + ⋯
L(s)  = 1  + 1.49·2-s + 0.577·3-s + 1.23·4-s − 1.50·5-s + 0.863·6-s − 0.0685·7-s + 0.357·8-s + 0.333·9-s − 2.25·10-s + 1.90·11-s + 0.715·12-s + 1.43·13-s − 0.102·14-s − 0.869·15-s − 0.704·16-s + 0.0645·17-s + 0.498·18-s − 0.264·19-s − 1.86·20-s − 0.0395·21-s + 2.85·22-s − 0.0348·23-s + 0.206·24-s + 1.26·25-s + 2.15·26-s + 0.192·27-s − 0.0848·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(55.2921\)
Root analytic conductor: \(7.43586\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(5.364319362\)
\(L(\frac12)\) \(\approx\) \(5.364319362\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 27T \)
59 \( 1 - 2.05e5T \)
good2 \( 1 - 16.9T + 128T^{2} \)
5 \( 1 + 421.T + 7.81e4T^{2} \)
7 \( 1 + 62.1T + 8.23e5T^{2} \)
11 \( 1 - 8.42e3T + 1.94e7T^{2} \)
13 \( 1 - 1.13e4T + 6.27e7T^{2} \)
17 \( 1 - 1.30e3T + 4.10e8T^{2} \)
19 \( 1 + 7.90e3T + 8.93e8T^{2} \)
23 \( 1 + 2.03e3T + 3.40e9T^{2} \)
29 \( 1 - 6.42e4T + 1.72e10T^{2} \)
31 \( 1 - 3.20e5T + 2.75e10T^{2} \)
37 \( 1 + 1.43e5T + 9.49e10T^{2} \)
41 \( 1 - 5.77e5T + 1.94e11T^{2} \)
43 \( 1 - 8.17e5T + 2.71e11T^{2} \)
47 \( 1 - 2.64e5T + 5.06e11T^{2} \)
53 \( 1 - 1.60e6T + 1.17e12T^{2} \)
61 \( 1 + 2.74e5T + 3.14e12T^{2} \)
67 \( 1 + 1.66e6T + 6.06e12T^{2} \)
71 \( 1 + 2.97e6T + 9.09e12T^{2} \)
73 \( 1 + 2.72e6T + 1.10e13T^{2} \)
79 \( 1 - 1.33e6T + 1.92e13T^{2} \)
83 \( 1 - 8.00e6T + 2.71e13T^{2} \)
89 \( 1 + 1.07e7T + 4.42e13T^{2} \)
97 \( 1 + 1.36e7T + 8.07e13T^{2} \)
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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

−11.80970607645915151892342100525, −10.92947272128943523334795201030, −9.120924221626273634647921422216, −8.276537798089252045139577753510, −6.94812961620447498816857818681, −6.10037214330510945344456968913, −4.21446175812451100199190882037, −4.04124893896306218653432141904, −2.97272317660500648993001718031, −1.06893488905343463139062254596, 1.06893488905343463139062254596, 2.97272317660500648993001718031, 4.04124893896306218653432141904, 4.21446175812451100199190882037, 6.10037214330510945344456968913, 6.94812961620447498816857818681, 8.276537798089252045139577753510, 9.120924221626273634647921422216, 10.92947272128943523334795201030, 11.80970607645915151892342100525

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