Properties

Label 2-177-1.1-c5-0-3
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $28.3879$
Root an. cond. $5.32803$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.10·2-s + 9·3-s − 27.5·4-s − 81.1·5-s − 18.9·6-s − 97.6·7-s + 125.·8-s + 81·9-s + 170.·10-s − 297.·11-s − 248.·12-s − 814.·13-s + 205.·14-s − 730.·15-s + 619.·16-s − 1.27e3·17-s − 170.·18-s + 801.·19-s + 2.23e3·20-s − 878.·21-s + 624.·22-s + 255.·23-s + 1.12e3·24-s + 3.46e3·25-s + 1.71e3·26-s + 729·27-s + 2.69e3·28-s + ⋯
L(s)  = 1  − 0.371·2-s + 0.577·3-s − 0.861·4-s − 1.45·5-s − 0.214·6-s − 0.753·7-s + 0.692·8-s + 0.333·9-s + 0.539·10-s − 0.740·11-s − 0.497·12-s − 1.33·13-s + 0.279·14-s − 0.838·15-s + 0.604·16-s − 1.07·17-s − 0.123·18-s + 0.509·19-s + 1.25·20-s − 0.434·21-s + 0.275·22-s + 0.100·23-s + 0.399·24-s + 1.10·25-s + 0.496·26-s + 0.192·27-s + 0.648·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(28.3879\)
Root analytic conductor: \(5.32803\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.5062249462\)
\(L(\frac12)\) \(\approx\) \(0.5062249462\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 9T \)
59 \( 1 + 3.48e3T \)
good2 \( 1 + 2.10T + 32T^{2} \)
5 \( 1 + 81.1T + 3.12e3T^{2} \)
7 \( 1 + 97.6T + 1.68e4T^{2} \)
11 \( 1 + 297.T + 1.61e5T^{2} \)
13 \( 1 + 814.T + 3.71e5T^{2} \)
17 \( 1 + 1.27e3T + 1.41e6T^{2} \)
19 \( 1 - 801.T + 2.47e6T^{2} \)
23 \( 1 - 255.T + 6.43e6T^{2} \)
29 \( 1 - 7.41e3T + 2.05e7T^{2} \)
31 \( 1 + 2.82e3T + 2.86e7T^{2} \)
37 \( 1 - 4.78e3T + 6.93e7T^{2} \)
41 \( 1 + 8.22e3T + 1.15e8T^{2} \)
43 \( 1 + 1.93e3T + 1.47e8T^{2} \)
47 \( 1 - 2.32e4T + 2.29e8T^{2} \)
53 \( 1 - 1.52e4T + 4.18e8T^{2} \)
61 \( 1 - 9.48e3T + 8.44e8T^{2} \)
67 \( 1 - 2.80e4T + 1.35e9T^{2} \)
71 \( 1 + 2.61e4T + 1.80e9T^{2} \)
73 \( 1 - 7.78e3T + 2.07e9T^{2} \)
79 \( 1 + 5.12e4T + 3.07e9T^{2} \)
83 \( 1 - 1.77e4T + 3.93e9T^{2} \)
89 \( 1 - 1.04e5T + 5.58e9T^{2} \)
97 \( 1 + 1.44e5T + 8.58e9T^{2} \)
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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

−11.95168517474090535329578456621, −10.57069367438870702784765497304, −9.661932842689821004699864135286, −8.679023574279218643182864968560, −7.83817282197910940236617186716, −7.02290036191451787405157943591, −4.96111322211336024105799360929, −4.02529843103957429161428081227, −2.78708458736810344394279393370, −0.44571073449456441806621265176, 0.44571073449456441806621265176, 2.78708458736810344394279393370, 4.02529843103957429161428081227, 4.96111322211336024105799360929, 7.02290036191451787405157943591, 7.83817282197910940236617186716, 8.679023574279218643182864968560, 9.661932842689821004699864135286, 10.57069367438870702784765497304, 11.95168517474090535329578456621

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