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Properties

Label 2-177-59.58-c4-0-24
Degree $2$
Conductor $177$
Sign $0.380 + 0.924i$
Analytic cond. $18.2964$
Root an. cond. $4.27743$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.44i·2-s + 5.19·3-s + 4.13·4-s − 16.2·5-s − 17.8i·6-s + 92.6·7-s − 69.3i·8-s + 27·9-s + 55.8i·10-s + 48.6i·11-s + 21.4·12-s + 216. i·13-s − 319. i·14-s − 84.2·15-s − 172.·16-s + 171.·17-s + ⋯
L(s)  = 1  − 0.861i·2-s + 0.577·3-s + 0.258·4-s − 0.648·5-s − 0.497i·6-s + 1.89·7-s − 1.08i·8-s + 0.333·9-s + 0.558i·10-s + 0.402i·11-s + 0.149·12-s + 1.28i·13-s − 1.62i·14-s − 0.374·15-s − 0.674·16-s + 0.595·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.380 + 0.924i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.380 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.380 + 0.924i$
Analytic conductor: \(18.2964\)
Root analytic conductor: \(4.27743\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :2),\ 0.380 + 0.924i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.951791422\)
\(L(\frac12)\) \(\approx\) \(2.951791422\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 5.19T \)
59 \( 1 + (1.32e3 + 3.21e3i)T \)
good2 \( 1 + 3.44iT - 16T^{2} \)
5 \( 1 + 16.2T + 625T^{2} \)
7 \( 1 - 92.6T + 2.40e3T^{2} \)
11 \( 1 - 48.6iT - 1.46e4T^{2} \)
13 \( 1 - 216. iT - 2.85e4T^{2} \)
17 \( 1 - 171.T + 8.35e4T^{2} \)
19 \( 1 - 267.T + 1.30e5T^{2} \)
23 \( 1 + 734. iT - 2.79e5T^{2} \)
29 \( 1 - 999.T + 7.07e5T^{2} \)
31 \( 1 - 568. iT - 9.23e5T^{2} \)
37 \( 1 + 1.00e3iT - 1.87e6T^{2} \)
41 \( 1 + 2.30e3T + 2.82e6T^{2} \)
43 \( 1 + 2.15e3iT - 3.41e6T^{2} \)
47 \( 1 - 1.19e3iT - 4.87e6T^{2} \)
53 \( 1 + 1.10e3T + 7.89e6T^{2} \)
61 \( 1 + 4.03e3iT - 1.38e7T^{2} \)
67 \( 1 - 8.14e3iT - 2.01e7T^{2} \)
71 \( 1 + 2.53e3T + 2.54e7T^{2} \)
73 \( 1 - 792. iT - 2.83e7T^{2} \)
79 \( 1 + 3.21e3T + 3.89e7T^{2} \)
83 \( 1 - 1.05e4iT - 4.74e7T^{2} \)
89 \( 1 - 226. iT - 6.27e7T^{2} \)
97 \( 1 - 7.28e3iT - 8.85e7T^{2} \)
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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

−11.82476878246842232938120063623, −11.03177108757163207335201421940, −10.05531934135324804116618361706, −8.743570801739397991353187844789, −7.81726455143429108531892661870, −6.87588082701924746806575716622, −4.82955825930004608710697116615, −3.87340998228927634232739615994, −2.30438911309978091747184993273, −1.30029760786146231123233349979, 1.44789491188125595659176555425, 3.13477294046757327043974937941, 4.80858751530189630704542926255, 5.74870158100831431997199258974, 7.52043497710961496882641536757, 7.85698380138212189678866266721, 8.558119221833397062801938525444, 10.26145974486809096769086047169, 11.42195818160967401764990038324, 11.88327838981620228917109730329

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