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Properties

Label 2-177-59.58-c4-0-18
Degree $2$
Conductor $177$
Sign $0.317 - 0.948i$
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  + 2.43i·2-s + 5.19·3-s + 10.0·4-s + 25.9·5-s + 12.6i·6-s − 17.4·7-s + 63.4i·8-s + 27·9-s + 63.0i·10-s + 44.7i·11-s + 52.4·12-s + 157. i·13-s − 42.4i·14-s + 134.·15-s + 7.22·16-s − 328.·17-s + ⋯
L(s)  = 1  + 0.607i·2-s + 0.577·3-s + 0.630·4-s + 1.03·5-s + 0.350i·6-s − 0.356·7-s + 0.991i·8-s + 0.333·9-s + 0.630i·10-s + 0.369i·11-s + 0.364·12-s + 0.931i·13-s − 0.216i·14-s + 0.598·15-s + 0.0282·16-s − 1.13·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.317 - 0.948i$
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: \((2,\ 177,\ (\ :2),\ 0.317 - 0.948i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.182550973\)
\(L(\frac12)\) \(\approx\) \(3.182550973\)
\(L(3)\) 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 \( 1 + (1.10e3 - 3.30e3i)T \)
good2 \( 1 - 2.43iT - 16T^{2} \)
5 \( 1 - 25.9T + 625T^{2} \)
7 \( 1 + 17.4T + 2.40e3T^{2} \)
11 \( 1 - 44.7iT - 1.46e4T^{2} \)
13 \( 1 - 157. iT - 2.85e4T^{2} \)
17 \( 1 + 328.T + 8.35e4T^{2} \)
19 \( 1 - 678.T + 1.30e5T^{2} \)
23 \( 1 + 424. iT - 2.79e5T^{2} \)
29 \( 1 - 870.T + 7.07e5T^{2} \)
31 \( 1 + 1.38e3iT - 9.23e5T^{2} \)
37 \( 1 - 2.56e3iT - 1.87e6T^{2} \)
41 \( 1 - 503.T + 2.82e6T^{2} \)
43 \( 1 + 2.00e3iT - 3.41e6T^{2} \)
47 \( 1 - 2.42e3iT - 4.87e6T^{2} \)
53 \( 1 + 1.52e3T + 7.89e6T^{2} \)
61 \( 1 + 5.70e3iT - 1.38e7T^{2} \)
67 \( 1 + 4.77e3iT - 2.01e7T^{2} \)
71 \( 1 + 2.79e3T + 2.54e7T^{2} \)
73 \( 1 + 4.71e3iT - 2.83e7T^{2} \)
79 \( 1 - 8.66e3T + 3.89e7T^{2} \)
83 \( 1 - 7.45e3iT - 4.74e7T^{2} \)
89 \( 1 + 4.00e3iT - 6.27e7T^{2} \)
97 \( 1 + 5.91e3iT - 8.85e7T^{2} \)
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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

−12.24709573290938060011630660254, −11.21598326505602273207121635101, −9.934506142736265513966605301297, −9.233614549206650673930198634530, −7.985334800865719286434114231165, −6.84507409144931327768848801005, −6.15321266553437875967873217580, −4.74871916196181677585677800640, −2.83769568787979979750792207660, −1.76477592992538165664031337366, 1.20250694368287816971032617320, 2.53216488356384332303225566456, 3.44339031969254939783032019802, 5.40676337747262340564384267769, 6.54881671613184720166545284655, 7.64541899078207669157688431745, 9.101826324900264228613504701347, 9.891798409196826283995386374751, 10.69835545975012440802966801596, 11.78986436278327670708616686515

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