Properties

Label 2-177-1.1-c7-0-21
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  + 3.80·2-s + 27·3-s − 113.·4-s − 87.8·5-s + 102.·6-s + 1.30e3·7-s − 918.·8-s + 729·9-s − 334.·10-s − 3.79e3·11-s − 3.06e3·12-s + 8.97e3·13-s + 4.96e3·14-s − 2.37e3·15-s + 1.10e4·16-s − 4.35e3·17-s + 2.77e3·18-s − 5.70e4·19-s + 9.97e3·20-s + 3.52e4·21-s − 1.44e4·22-s + 9.66e4·23-s − 2.47e4·24-s − 7.04e4·25-s + 3.41e4·26-s + 1.96e4·27-s − 1.48e5·28-s + ⋯
L(s)  = 1  + 0.335·2-s + 0.577·3-s − 0.887·4-s − 0.314·5-s + 0.193·6-s + 1.43·7-s − 0.634·8-s + 0.333·9-s − 0.105·10-s − 0.860·11-s − 0.512·12-s + 1.13·13-s + 0.483·14-s − 0.181·15-s + 0.674·16-s − 0.215·17-s + 0.111·18-s − 1.90·19-s + 0.278·20-s + 0.830·21-s − 0.289·22-s + 1.65·23-s − 0.366·24-s − 0.901·25-s + 0.380·26-s + 0.192·27-s − 1.27·28-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

\(L(4)\) \(\approx\) \(2.618684343\)
\(L(\frac12)\) \(\approx\) \(2.618684343\)
\(L(\frac{9}{2})\) 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 \( 1 - 3.80T + 128T^{2} \)
5 \( 1 + 87.8T + 7.81e4T^{2} \)
7 \( 1 - 1.30e3T + 8.23e5T^{2} \)
11 \( 1 + 3.79e3T + 1.94e7T^{2} \)
13 \( 1 - 8.97e3T + 6.27e7T^{2} \)
17 \( 1 + 4.35e3T + 4.10e8T^{2} \)
19 \( 1 + 5.70e4T + 8.93e8T^{2} \)
23 \( 1 - 9.66e4T + 3.40e9T^{2} \)
29 \( 1 - 2.27e4T + 1.72e10T^{2} \)
31 \( 1 + 5.33e4T + 2.75e10T^{2} \)
37 \( 1 - 3.19e5T + 9.49e10T^{2} \)
41 \( 1 - 4.76e5T + 1.94e11T^{2} \)
43 \( 1 - 7.07e5T + 2.71e11T^{2} \)
47 \( 1 - 9.46e5T + 5.06e11T^{2} \)
53 \( 1 - 7.99e5T + 1.17e12T^{2} \)
61 \( 1 - 5.39e5T + 3.14e12T^{2} \)
67 \( 1 - 2.90e6T + 6.06e12T^{2} \)
71 \( 1 + 5.23e5T + 9.09e12T^{2} \)
73 \( 1 + 2.11e6T + 1.10e13T^{2} \)
79 \( 1 + 3.80e6T + 1.92e13T^{2} \)
83 \( 1 - 1.88e6T + 2.71e13T^{2} \)
89 \( 1 - 6.48e6T + 4.42e13T^{2} \)
97 \( 1 - 2.84e6T + 8.07e13T^{2} \)
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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

−11.27460207108616520821328303368, −10.56912398835680510994797515547, −9.004815143109664119247358709215, −8.439199245855423557218390397148, −7.59614226165164540264077980907, −5.85771071490862026034525451435, −4.66135290797146064576609097959, −3.93926098006400583832085341726, −2.37898631662826613415427442928, −0.851874011131207138412055655842, 0.851874011131207138412055655842, 2.37898631662826613415427442928, 3.93926098006400583832085341726, 4.66135290797146064576609097959, 5.85771071490862026034525451435, 7.59614226165164540264077980907, 8.439199245855423557218390397148, 9.004815143109664119247358709215, 10.56912398835680510994797515547, 11.27460207108616520821328303368

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