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Properties

Label 2-177-3.2-c4-0-11
Degree $2$
Conductor $177$
Sign $0.731 + 0.681i$
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  − 7.49i·2-s + (−6.13 + 6.58i)3-s − 40.1·4-s − 24.9i·5-s + (49.3 + 45.9i)6-s − 52.3·7-s + 181. i·8-s + (−5.78 − 80.7i)9-s − 187.·10-s + 127. i·11-s + (246. − 264. i)12-s − 56.9·13-s + 392. i·14-s + (164. + 153. i)15-s + 715.·16-s + 227. i·17-s + ⋯
L(s)  = 1  − 1.87i·2-s + (−0.681 + 0.731i)3-s − 2.51·4-s − 0.998i·5-s + (1.37 + 1.27i)6-s − 1.06·7-s + 2.83i·8-s + (−0.0714 − 0.997i)9-s − 1.87·10-s + 1.05i·11-s + (1.71 − 1.83i)12-s − 0.336·13-s + 2.00i·14-s + (0.730 + 0.680i)15-s + 2.79·16-s + 0.787i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 + 0.681i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.731 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.731 + 0.681i$
Analytic conductor: \(18.2964\)
Root analytic conductor: \(4.27743\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :2),\ 0.731 + 0.681i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.5375927772\)
\(L(\frac12)\) \(\approx\) \(0.5375927772\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (6.13 - 6.58i)T \)
59 \( 1 + 453. iT \)
good2 \( 1 + 7.49iT - 16T^{2} \)
5 \( 1 + 24.9iT - 625T^{2} \)
7 \( 1 + 52.3T + 2.40e3T^{2} \)
11 \( 1 - 127. iT - 1.46e4T^{2} \)
13 \( 1 + 56.9T + 2.85e4T^{2} \)
17 \( 1 - 227. iT - 8.35e4T^{2} \)
19 \( 1 + 187.T + 1.30e5T^{2} \)
23 \( 1 + 905. iT - 2.79e5T^{2} \)
29 \( 1 + 1.37e3iT - 7.07e5T^{2} \)
31 \( 1 - 812.T + 9.23e5T^{2} \)
37 \( 1 - 253.T + 1.87e6T^{2} \)
41 \( 1 - 1.74e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.42e3T + 3.41e6T^{2} \)
47 \( 1 - 1.93e3iT - 4.87e6T^{2} \)
53 \( 1 - 4.44e3iT - 7.89e6T^{2} \)
61 \( 1 + 2.33e3T + 1.38e7T^{2} \)
67 \( 1 - 4.91e3T + 2.01e7T^{2} \)
71 \( 1 - 8.30e3iT - 2.54e7T^{2} \)
73 \( 1 + 3.76e3T + 2.83e7T^{2} \)
79 \( 1 - 2.12e3T + 3.89e7T^{2} \)
83 \( 1 - 7.49e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.09e4iT - 6.27e7T^{2} \)
97 \( 1 + 1.60e4T + 8.85e7T^{2} \)
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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

−12.04327894511667774823403841134, −10.82640604507798897986210808972, −9.980570455166462751963093842096, −9.494162136907735043611580472830, −8.476070026303371925350385716390, −6.17890489819856986317557544334, −4.63914046007390837704424753400, −4.20009117469794521824124961786, −2.61085336833128773645509918226, −0.848591428864008707687791018110, 0.31951462648099470928865114616, 3.31575876932044332012300976670, 5.20020305865992040862532699760, 6.11103449486995938914659279024, 6.85205852096258570197365299552, 7.44996682956923615498854920267, 8.708749321283445297368395426064, 9.891055457762206715817185108429, 11.10312864820820941151766508372, 12.50212332491496482165773849355

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