Properties

Label 2-177-59.58-c4-0-16
Degree $2$
Conductor $177$
Sign $0.586 - 0.810i$
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  + 6.10i·2-s − 5.19·3-s − 21.2·4-s − 12.8·5-s − 31.7i·6-s + 4.61·7-s − 32.0i·8-s + 27·9-s − 78.4i·10-s − 159. i·11-s + 110.·12-s − 148. i·13-s + 28.1i·14-s + 66.7·15-s − 144.·16-s + 154.·17-s + ⋯
L(s)  = 1  + 1.52i·2-s − 0.577·3-s − 1.32·4-s − 0.513·5-s − 0.880i·6-s + 0.0942·7-s − 0.500i·8-s + 0.333·9-s − 0.784i·10-s − 1.31i·11-s + 0.766·12-s − 0.881i·13-s + 0.143i·14-s + 0.296·15-s − 0.564·16-s + 0.534·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 - 0.810i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.586 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.586 - 0.810i$
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: \((2,\ 177,\ (\ :2),\ 0.586 - 0.810i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.139575311\)
\(L(\frac12)\) \(\approx\) \(1.139575311\)
\(L(3)\) 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 \( 1 + (-2.04e3 + 2.82e3i)T \)
good2 \( 1 - 6.10iT - 16T^{2} \)
5 \( 1 + 12.8T + 625T^{2} \)
7 \( 1 - 4.61T + 2.40e3T^{2} \)
11 \( 1 + 159. iT - 1.46e4T^{2} \)
13 \( 1 + 148. iT - 2.85e4T^{2} \)
17 \( 1 - 154.T + 8.35e4T^{2} \)
19 \( 1 - 313.T + 1.30e5T^{2} \)
23 \( 1 - 420. iT - 2.79e5T^{2} \)
29 \( 1 - 1.42e3T + 7.07e5T^{2} \)
31 \( 1 - 1.10e3iT - 9.23e5T^{2} \)
37 \( 1 - 186. iT - 1.87e6T^{2} \)
41 \( 1 - 435.T + 2.82e6T^{2} \)
43 \( 1 + 2.36e3iT - 3.41e6T^{2} \)
47 \( 1 + 2.36e3iT - 4.87e6T^{2} \)
53 \( 1 - 3.94e3T + 7.89e6T^{2} \)
61 \( 1 + 1.73e3iT - 1.38e7T^{2} \)
67 \( 1 - 2.71e3iT - 2.01e7T^{2} \)
71 \( 1 - 5.43e3T + 2.54e7T^{2} \)
73 \( 1 - 2.07e3iT - 2.83e7T^{2} \)
79 \( 1 - 1.07e4T + 3.89e7T^{2} \)
83 \( 1 + 1.09e4iT - 4.74e7T^{2} \)
89 \( 1 + 1.73e3iT - 6.27e7T^{2} \)
97 \( 1 + 3.59e3iT - 8.85e7T^{2} \)
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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

−12.12809438498601468719672712381, −11.26443560765922028742004605654, −10.07155941139016971009710965898, −8.609212448971439485106048231402, −7.909374438236446576822948963940, −6.90857283271205872230412673606, −5.78014233754393167843932096527, −5.11820206451958899679763640337, −3.47012638101266496076157841417, −0.59756065813692034207403646600, 1.02513787021111082251684641593, 2.40534371194816987493166760142, 3.98987761799690426641290555889, 4.80590090537965240032745041944, 6.59302447388165709798380864068, 7.81959114815346272351341120890, 9.405616563633726891800823215219, 10.03578992808756046134689044923, 11.07254225286977963556452797506, 11.95677672030056523457900133409

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