Properties

Label 2-177-1.1-c9-0-70
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $91.1613$
Root an. cond. $9.54784$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 29.8·2-s + 81·3-s + 376.·4-s + 1.19e3·5-s − 2.41e3·6-s + 3.75e3·7-s + 4.03e3·8-s + 6.56e3·9-s − 3.57e4·10-s + 5.75e4·11-s + 3.05e4·12-s − 9.66e4·13-s − 1.11e5·14-s + 9.70e4·15-s − 3.13e5·16-s + 1.99e5·17-s − 1.95e5·18-s − 3.41e5·19-s + 4.51e5·20-s + 3.04e5·21-s − 1.71e6·22-s − 7.30e5·23-s + 3.26e5·24-s − 5.17e5·25-s + 2.88e6·26-s + 5.31e5·27-s + 1.41e6·28-s + ⋯
L(s)  = 1  − 1.31·2-s + 0.577·3-s + 0.735·4-s + 0.857·5-s − 0.760·6-s + 0.590·7-s + 0.348·8-s + 0.333·9-s − 1.12·10-s + 1.18·11-s + 0.424·12-s − 0.938·13-s − 0.778·14-s + 0.494·15-s − 1.19·16-s + 0.579·17-s − 0.439·18-s − 0.601·19-s + 0.630·20-s + 0.341·21-s − 1.56·22-s − 0.543·23-s + 0.200·24-s − 0.264·25-s + 1.23·26-s + 0.192·27-s + 0.434·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(91.1613\)
Root analytic conductor: \(9.54784\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 81T \)
59 \( 1 - 1.21e7T \)
good2 \( 1 + 29.8T + 512T^{2} \)
5 \( 1 - 1.19e3T + 1.95e6T^{2} \)
7 \( 1 - 3.75e3T + 4.03e7T^{2} \)
11 \( 1 - 5.75e4T + 2.35e9T^{2} \)
13 \( 1 + 9.66e4T + 1.06e10T^{2} \)
17 \( 1 - 1.99e5T + 1.18e11T^{2} \)
19 \( 1 + 3.41e5T + 3.22e11T^{2} \)
23 \( 1 + 7.30e5T + 1.80e12T^{2} \)
29 \( 1 + 6.39e6T + 1.45e13T^{2} \)
31 \( 1 + 5.53e6T + 2.64e13T^{2} \)
37 \( 1 + 1.72e7T + 1.29e14T^{2} \)
41 \( 1 + 3.16e7T + 3.27e14T^{2} \)
43 \( 1 - 1.31e7T + 5.02e14T^{2} \)
47 \( 1 + 1.21e7T + 1.11e15T^{2} \)
53 \( 1 - 3.91e7T + 3.29e15T^{2} \)
61 \( 1 + 4.88e7T + 1.16e16T^{2} \)
67 \( 1 - 4.81e7T + 2.72e16T^{2} \)
71 \( 1 + 1.07e7T + 4.58e16T^{2} \)
73 \( 1 - 3.67e8T + 5.88e16T^{2} \)
79 \( 1 - 3.81e8T + 1.19e17T^{2} \)
83 \( 1 + 3.92e8T + 1.86e17T^{2} \)
89 \( 1 + 2.06e8T + 3.50e17T^{2} \)
97 \( 1 + 1.55e9T + 7.60e17T^{2} \)
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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

−10.06521830022828301095210751289, −9.470101318060336148252053232570, −8.695089195749736091451864595209, −7.70229938084572138308679434762, −6.77009082509127170468742350664, −5.27380419166635753022457872192, −3.84095033925974597554131577366, −2.00792244759860698426982461456, −1.56675333517052479529369809197, 0, 1.56675333517052479529369809197, 2.00792244759860698426982461456, 3.84095033925974597554131577366, 5.27380419166635753022457872192, 6.77009082509127170468742350664, 7.70229938084572138308679434762, 8.695089195749736091451864595209, 9.470101318060336148252053232570, 10.06521830022828301095210751289

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