Properties

Label 2-177-59.58-c4-0-5
Degree $2$
Conductor $177$
Sign $-0.345 - 0.938i$
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  − 0.850i·2-s − 5.19·3-s + 15.2·4-s − 11.2·5-s + 4.41i·6-s − 26.4·7-s − 26.5i·8-s + 27·9-s + 9.58i·10-s − 19.9i·11-s − 79.3·12-s + 190. i·13-s + 22.4i·14-s + 58.5·15-s + 221.·16-s − 159.·17-s + ⋯
L(s)  = 1  − 0.212i·2-s − 0.577·3-s + 0.954·4-s − 0.450·5-s + 0.122i·6-s − 0.539·7-s − 0.415i·8-s + 0.333·9-s + 0.0958i·10-s − 0.165i·11-s − 0.551·12-s + 1.12i·13-s + 0.114i·14-s + 0.260·15-s + 0.866·16-s − 0.551·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.345 - 0.938i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.345 - 0.938i$
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: \((2,\ 177,\ (\ :2),\ -0.345 - 0.938i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.7817990821\)
\(L(\frac12)\) \(\approx\) \(0.7817990821\)
\(L(3)\) 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 \( 1 + (1.20e3 + 3.26e3i)T \)
good2 \( 1 + 0.850iT - 16T^{2} \)
5 \( 1 + 11.2T + 625T^{2} \)
7 \( 1 + 26.4T + 2.40e3T^{2} \)
11 \( 1 + 19.9iT - 1.46e4T^{2} \)
13 \( 1 - 190. iT - 2.85e4T^{2} \)
17 \( 1 + 159.T + 8.35e4T^{2} \)
19 \( 1 + 294.T + 1.30e5T^{2} \)
23 \( 1 - 165. iT - 2.79e5T^{2} \)
29 \( 1 - 513.T + 7.07e5T^{2} \)
31 \( 1 - 1.62e3iT - 9.23e5T^{2} \)
37 \( 1 - 1.77e3iT - 1.87e6T^{2} \)
41 \( 1 + 2.06e3T + 2.82e6T^{2} \)
43 \( 1 - 2.18e3iT - 3.41e6T^{2} \)
47 \( 1 - 4.25e3iT - 4.87e6T^{2} \)
53 \( 1 - 246.T + 7.89e6T^{2} \)
61 \( 1 + 1.49e3iT - 1.38e7T^{2} \)
67 \( 1 + 3.88e3iT - 2.01e7T^{2} \)
71 \( 1 - 1.86e3T + 2.54e7T^{2} \)
73 \( 1 - 619. iT - 2.83e7T^{2} \)
79 \( 1 - 2.37e3T + 3.89e7T^{2} \)
83 \( 1 + 4.60e3iT - 4.74e7T^{2} \)
89 \( 1 + 8.91e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.37e3iT - 8.85e7T^{2} \)
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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

−12.07307201785154505093644119570, −11.40832370188374128900655135983, −10.57956572355885774744047062725, −9.512574809194706727708100997786, −8.112337048168993951058930851795, −6.78191967199364188055043599923, −6.31340256057755032067247393472, −4.61878314838349339448227407046, −3.22548575748504191903866118948, −1.61964668909331041654599999937, 0.29071644494496947263790160418, 2.29957382935338180604031963879, 3.84240430406884883214945838288, 5.47617112410286644519821037208, 6.43140794175943665057214102228, 7.37222005076413838360990828173, 8.413180284227111881655125835539, 10.01583322190417147929103283905, 10.77281558014440253233379893234, 11.71406291209168216648691258267

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