Properties

Label 2-177-1.1-c3-0-25
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $10.4433$
Root an. cond. $3.23161$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.74·2-s − 3·3-s − 0.442·4-s + 13.7·5-s − 8.24·6-s − 35.6·7-s − 23.2·8-s + 9·9-s + 37.8·10-s − 30.8·11-s + 1.32·12-s + 14.6·13-s − 97.9·14-s − 41.3·15-s − 60.2·16-s − 77.8·17-s + 24.7·18-s + 98.6·19-s − 6.09·20-s + 106.·21-s − 84.8·22-s − 117.·23-s + 69.6·24-s + 64.7·25-s + 40.2·26-s − 27·27-s + 15.7·28-s + ⋯
L(s)  = 1  + 0.971·2-s − 0.577·3-s − 0.0552·4-s + 1.23·5-s − 0.561·6-s − 1.92·7-s − 1.02·8-s + 0.333·9-s + 1.19·10-s − 0.845·11-s + 0.0319·12-s + 0.312·13-s − 1.87·14-s − 0.711·15-s − 0.941·16-s − 1.11·17-s + 0.323·18-s + 1.19·19-s − 0.0680·20-s + 1.11·21-s − 0.822·22-s − 1.06·23-s + 0.592·24-s + 0.517·25-s + 0.303·26-s − 0.192·27-s + 0.106·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(10.4433\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
59 \( 1 - 59T \)
good2 \( 1 - 2.74T + 8T^{2} \)
5 \( 1 - 13.7T + 125T^{2} \)
7 \( 1 + 35.6T + 343T^{2} \)
11 \( 1 + 30.8T + 1.33e3T^{2} \)
13 \( 1 - 14.6T + 2.19e3T^{2} \)
17 \( 1 + 77.8T + 4.91e3T^{2} \)
19 \( 1 - 98.6T + 6.85e3T^{2} \)
23 \( 1 + 117.T + 1.21e4T^{2} \)
29 \( 1 + 54.2T + 2.43e4T^{2} \)
31 \( 1 + 231.T + 2.97e4T^{2} \)
37 \( 1 + 258.T + 5.06e4T^{2} \)
41 \( 1 - 326.T + 6.89e4T^{2} \)
43 \( 1 - 250.T + 7.95e4T^{2} \)
47 \( 1 - 509.T + 1.03e5T^{2} \)
53 \( 1 + 313.T + 1.48e5T^{2} \)
61 \( 1 - 528.T + 2.26e5T^{2} \)
67 \( 1 + 373.T + 3.00e5T^{2} \)
71 \( 1 + 521.T + 3.57e5T^{2} \)
73 \( 1 - 809.T + 3.89e5T^{2} \)
79 \( 1 + 926.T + 4.93e5T^{2} \)
83 \( 1 + 1.31e3T + 5.71e5T^{2} \)
89 \( 1 - 510.T + 7.04e5T^{2} \)
97 \( 1 + 856.T + 9.12e5T^{2} \)
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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

−12.19516919423652906773164715130, −10.70795976152786335727121003603, −9.703363315624439242214745676766, −9.134756169955798200003650325293, −7.00639941377411189523971430672, −5.91820165093941836294734707383, −5.56035653165172438333234127124, −3.91013703573279453567969678975, −2.60567859680536277581454774684, 0, 2.60567859680536277581454774684, 3.91013703573279453567969678975, 5.56035653165172438333234127124, 5.91820165093941836294734707383, 7.00639941377411189523971430672, 9.134756169955798200003650325293, 9.703363315624439242214745676766, 10.70795976152786335727121003603, 12.19516919423652906773164715130

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