Properties

Label 2-177-1.1-c7-0-59
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 13.0·2-s − 27·3-s + 41.1·4-s + 167.·5-s − 351.·6-s + 887.·7-s − 1.13e3·8-s + 729·9-s + 2.17e3·10-s − 7.78e3·11-s − 1.10e3·12-s + 4.60e3·13-s + 1.15e4·14-s − 4.51e3·15-s − 1.99e4·16-s − 6.05e3·17-s + 9.47e3·18-s + 1.35e4·19-s + 6.87e3·20-s − 2.39e4·21-s − 1.01e5·22-s + 3.52e4·23-s + 3.05e4·24-s − 5.01e4·25-s + 5.99e4·26-s − 1.96e4·27-s + 3.64e4·28-s + ⋯
L(s)  = 1  + 1.14·2-s − 0.577·3-s + 0.321·4-s + 0.598·5-s − 0.663·6-s + 0.977·7-s − 0.780·8-s + 0.333·9-s + 0.687·10-s − 1.76·11-s − 0.185·12-s + 0.581·13-s + 1.12·14-s − 0.345·15-s − 1.21·16-s − 0.298·17-s + 0.383·18-s + 0.453·19-s + 0.192·20-s − 0.564·21-s − 2.02·22-s + 0.604·23-s + 0.450·24-s − 0.641·25-s + 0.668·26-s − 0.192·27-s + 0.313·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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 \( 1 - 13.0T + 128T^{2} \)
5 \( 1 - 167.T + 7.81e4T^{2} \)
7 \( 1 - 887.T + 8.23e5T^{2} \)
11 \( 1 + 7.78e3T + 1.94e7T^{2} \)
13 \( 1 - 4.60e3T + 6.27e7T^{2} \)
17 \( 1 + 6.05e3T + 4.10e8T^{2} \)
19 \( 1 - 1.35e4T + 8.93e8T^{2} \)
23 \( 1 - 3.52e4T + 3.40e9T^{2} \)
29 \( 1 - 7.83e4T + 1.72e10T^{2} \)
31 \( 1 + 1.70e5T + 2.75e10T^{2} \)
37 \( 1 + 2.63e5T + 9.49e10T^{2} \)
41 \( 1 + 1.98e5T + 1.94e11T^{2} \)
43 \( 1 + 4.44e5T + 2.71e11T^{2} \)
47 \( 1 + 5.31e5T + 5.06e11T^{2} \)
53 \( 1 + 1.00e6T + 1.17e12T^{2} \)
61 \( 1 + 1.77e6T + 3.14e12T^{2} \)
67 \( 1 + 2.37e6T + 6.06e12T^{2} \)
71 \( 1 + 5.87e6T + 9.09e12T^{2} \)
73 \( 1 + 2.90e6T + 1.10e13T^{2} \)
79 \( 1 + 1.66e6T + 1.92e13T^{2} \)
83 \( 1 - 8.18e6T + 2.71e13T^{2} \)
89 \( 1 - 1.13e7T + 4.42e13T^{2} \)
97 \( 1 - 2.43e5T + 8.07e13T^{2} \)
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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

−11.09982804502068330048081301107, −10.21490348764555449053828058791, −8.828305460416954273068638644758, −7.61880872109579581776510525867, −6.18364832931992730942427805891, −5.26827615001935151750828759923, −4.74652432992454817465707469429, −3.18196013146654100597708254876, −1.79272410444389339094485643461, 0, 1.79272410444389339094485643461, 3.18196013146654100597708254876, 4.74652432992454817465707469429, 5.26827615001935151750828759923, 6.18364832931992730942427805891, 7.61880872109579581776510525867, 8.828305460416954273068638644758, 10.21490348764555449053828058791, 11.09982804502068330048081301107

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