Properties

Label 2-177-59.58-c4-0-26
Degree $2$
Conductor $177$
Sign $0.867 - 0.497i$
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  + 1.07i·2-s + 5.19·3-s + 14.8·4-s + 26.1·5-s + 5.60i·6-s + 49.1·7-s + 33.2i·8-s + 27·9-s + 28.1i·10-s + 190. i·11-s + 77.1·12-s − 198. i·13-s + 52.9i·14-s + 135.·15-s + 201.·16-s − 81.6·17-s + ⋯
L(s)  = 1  + 0.269i·2-s + 0.577·3-s + 0.927·4-s + 1.04·5-s + 0.155i·6-s + 1.00·7-s + 0.519i·8-s + 0.333·9-s + 0.281i·10-s + 1.57i·11-s + 0.535·12-s − 1.17i·13-s + 0.270i·14-s + 0.603·15-s + 0.787·16-s − 0.282·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.867 - 0.497i$
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: \((2,\ 177,\ (\ :2),\ 0.867 - 0.497i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.625097332\)
\(L(\frac12)\) \(\approx\) \(3.625097332\)
\(L(3)\) 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 \( 1 + (3.01e3 - 1.73e3i)T \)
good2 \( 1 - 1.07iT - 16T^{2} \)
5 \( 1 - 26.1T + 625T^{2} \)
7 \( 1 - 49.1T + 2.40e3T^{2} \)
11 \( 1 - 190. iT - 1.46e4T^{2} \)
13 \( 1 + 198. iT - 2.85e4T^{2} \)
17 \( 1 + 81.6T + 8.35e4T^{2} \)
19 \( 1 + 590.T + 1.30e5T^{2} \)
23 \( 1 + 808. iT - 2.79e5T^{2} \)
29 \( 1 + 714.T + 7.07e5T^{2} \)
31 \( 1 - 1.37e3iT - 9.23e5T^{2} \)
37 \( 1 + 2.10e3iT - 1.87e6T^{2} \)
41 \( 1 - 1.13e3T + 2.82e6T^{2} \)
43 \( 1 + 349. iT - 3.41e6T^{2} \)
47 \( 1 - 2.25e3iT - 4.87e6T^{2} \)
53 \( 1 - 4.73e3T + 7.89e6T^{2} \)
61 \( 1 - 1.50e3iT - 1.38e7T^{2} \)
67 \( 1 - 233. iT - 2.01e7T^{2} \)
71 \( 1 - 2.66e3T + 2.54e7T^{2} \)
73 \( 1 + 9.12e3iT - 2.83e7T^{2} \)
79 \( 1 + 8.28e3T + 3.89e7T^{2} \)
83 \( 1 + 7.47e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.05e4iT - 6.27e7T^{2} \)
97 \( 1 + 1.50e4iT - 8.85e7T^{2} \)
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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

−12.34579329463157173551241664990, −10.74666692890276938892965208439, −10.33135201350625146198255620921, −8.941071318423947646256667578570, −7.891709922467858209473814073572, −6.95404594863031986634976564172, −5.78565094697021556822777578425, −4.53190175389419617862373618726, −2.45084226348020311308212278317, −1.79220830373544717316347366631, 1.53832079695837321375646877806, 2.35141874523317527858375546053, 3.90907018775507364427667517538, 5.66599312989097717050117810705, 6.58254849682825496199785153086, 7.917130946834949753874673317050, 8.909609845150393104874211900428, 9.970629855433907439204574008736, 11.16458817043536273288676534692, 11.54231806372425622935306819372

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