Properties

Label 2-177-1.1-c7-0-64
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  + 16.2·2-s + 27·3-s + 136.·4-s − 30.6·5-s + 439.·6-s − 1.35e3·7-s + 137.·8-s + 729·9-s − 498.·10-s + 1.82e3·11-s + 3.68e3·12-s − 1.17e3·13-s − 2.20e4·14-s − 828.·15-s − 1.52e4·16-s + 1.31e4·17-s + 1.18e4·18-s − 1.85e4·19-s − 4.18e3·20-s − 3.66e4·21-s + 2.96e4·22-s − 1.45e4·23-s + 3.71e3·24-s − 7.71e4·25-s − 1.90e4·26-s + 1.96e4·27-s − 1.85e5·28-s + ⋯
L(s)  = 1  + 1.43·2-s + 0.577·3-s + 1.06·4-s − 0.109·5-s + 0.829·6-s − 1.49·7-s + 0.0948·8-s + 0.333·9-s − 0.157·10-s + 0.413·11-s + 0.615·12-s − 0.148·13-s − 2.14·14-s − 0.0633·15-s − 0.929·16-s + 0.650·17-s + 0.479·18-s − 0.621·19-s − 0.116·20-s − 0.863·21-s + 0.594·22-s − 0.249·23-s + 0.0547·24-s − 0.987·25-s − 0.212·26-s + 0.192·27-s − 1.59·28-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 \( 1 - 16.2T + 128T^{2} \)
5 \( 1 + 30.6T + 7.81e4T^{2} \)
7 \( 1 + 1.35e3T + 8.23e5T^{2} \)
11 \( 1 - 1.82e3T + 1.94e7T^{2} \)
13 \( 1 + 1.17e3T + 6.27e7T^{2} \)
17 \( 1 - 1.31e4T + 4.10e8T^{2} \)
19 \( 1 + 1.85e4T + 8.93e8T^{2} \)
23 \( 1 + 1.45e4T + 3.40e9T^{2} \)
29 \( 1 + 1.83e5T + 1.72e10T^{2} \)
31 \( 1 + 2.65e5T + 2.75e10T^{2} \)
37 \( 1 + 2.60e5T + 9.49e10T^{2} \)
41 \( 1 - 7.19e5T + 1.94e11T^{2} \)
43 \( 1 + 7.07e5T + 2.71e11T^{2} \)
47 \( 1 - 1.30e6T + 5.06e11T^{2} \)
53 \( 1 + 1.80e6T + 1.17e12T^{2} \)
61 \( 1 - 1.38e6T + 3.14e12T^{2} \)
67 \( 1 - 2.41e6T + 6.06e12T^{2} \)
71 \( 1 - 2.72e6T + 9.09e12T^{2} \)
73 \( 1 - 3.97e6T + 1.10e13T^{2} \)
79 \( 1 + 6.91e6T + 1.92e13T^{2} \)
83 \( 1 - 8.32e6T + 2.71e13T^{2} \)
89 \( 1 + 6.71e6T + 4.42e13T^{2} \)
97 \( 1 + 4.77e6T + 8.07e13T^{2} \)
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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

−11.19166647160924987294990017239, −9.794765061088995776046229229792, −9.041923290540610470214229499600, −7.45018063363387845024241021243, −6.41542881747670883192379756482, −5.50289762625475951641870914506, −3.92623280529556797033940535543, −3.44932486333415949416586782098, −2.17546885803636557412165301784, 0, 2.17546885803636557412165301784, 3.44932486333415949416586782098, 3.92623280529556797033940535543, 5.50289762625475951641870914506, 6.41542881747670883192379756482, 7.45018063363387845024241021243, 9.041923290540610470214229499600, 9.794765061088995776046229229792, 11.19166647160924987294990017239

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