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Properties

Label 2-177-177.176-c3-0-46
Degree $2$
Conductor $177$
Sign $-0.350 - 0.936i$
Analytic cond. $10.4433$
Root an. cond. $3.23161$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.56·2-s + (−1.93 − 4.82i)3-s + 22.9·4-s − 10.3i·5-s + (10.7 + 26.8i)6-s − 5.13·7-s − 83.3·8-s + (−19.5 + 18.6i)9-s + 57.4i·10-s + 8.23·11-s + (−44.4 − 110. i)12-s − 53.3i·13-s + 28.5·14-s + (−49.7 + 19.9i)15-s + 280.·16-s − 93.0i·17-s + ⋯
L(s)  = 1  − 1.96·2-s + (−0.372 − 0.928i)3-s + 2.87·4-s − 0.922i·5-s + (0.732 + 1.82i)6-s − 0.277·7-s − 3.68·8-s + (−0.722 + 0.690i)9-s + 1.81i·10-s + 0.225·11-s + (−1.06 − 2.66i)12-s − 1.13i·13-s + 0.545·14-s + (−0.856 + 0.343i)15-s + 4.37·16-s − 1.32i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.350 - 0.936i$
Analytic conductor: \(10.4433\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ -0.350 - 0.936i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0703798 + 0.101542i\)
\(L(\frac12)\) \(\approx\) \(0.0703798 + 0.101542i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.93 + 4.82i)T \)
59 \( 1 + (-453. - 10.2i)T \)
good2 \( 1 + 5.56T + 8T^{2} \)
5 \( 1 + 10.3iT - 125T^{2} \)
7 \( 1 + 5.13T + 343T^{2} \)
11 \( 1 - 8.23T + 1.33e3T^{2} \)
13 \( 1 + 53.3iT - 2.19e3T^{2} \)
17 \( 1 + 93.0iT - 4.91e3T^{2} \)
19 \( 1 + 84.1T + 6.85e3T^{2} \)
23 \( 1 + 138.T + 1.21e4T^{2} \)
29 \( 1 - 249. iT - 2.43e4T^{2} \)
31 \( 1 + 203. iT - 2.97e4T^{2} \)
37 \( 1 + 60.4iT - 5.06e4T^{2} \)
41 \( 1 + 36.5iT - 6.89e4T^{2} \)
43 \( 1 - 368. iT - 7.95e4T^{2} \)
47 \( 1 - 112.T + 1.03e5T^{2} \)
53 \( 1 - 318. iT - 1.48e5T^{2} \)
61 \( 1 - 214. iT - 2.26e5T^{2} \)
67 \( 1 - 309. iT - 3.00e5T^{2} \)
71 \( 1 - 830. iT - 3.57e5T^{2} \)
73 \( 1 + 110. iT - 3.89e5T^{2} \)
79 \( 1 + 180.T + 4.93e5T^{2} \)
83 \( 1 - 339.T + 5.71e5T^{2} \)
89 \( 1 - 743.T + 7.04e5T^{2} \)
97 \( 1 - 439. iT - 9.12e5T^{2} \)
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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

−11.37986929039296235063157394283, −10.41619199429850121597198167895, −9.327285014206687473916246400583, −8.432943840450108215463848184016, −7.68823972605582097193769273644, −6.66369854696288752652161308022, −5.58159468406267030787184585759, −2.61247594141974562858964773891, −1.16394757749461894433503979414, −0.12053877137041534408445852719, 2.11828596884786825128214971039, 3.70351005652677918707784322208, 6.21111146478973291182009268381, 6.65889759919620734470258234477, 8.161385193598203891849219321195, 9.043422629525832341123657633479, 10.05195116632916926629614645177, 10.53111398808231418477216822242, 11.38934642842263276678011093004, 12.19457241870740422985533542987

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