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Properties

Label 2-177-3.2-c4-0-2
Degree $2$
Conductor $177$
Sign $-0.719 - 0.694i$
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  − 5.57i·2-s + (−6.24 + 6.47i)3-s − 15.1·4-s + 38.3i·5-s + (36.1 + 34.8i)6-s + 12.9·7-s − 4.88i·8-s + (−2.93 − 80.9i)9-s + 213.·10-s + 6.78i·11-s + (94.4 − 97.9i)12-s + 101.·13-s − 72.4i·14-s + (−248. − 239. i)15-s − 269.·16-s + 232. i·17-s + ⋯
L(s)  = 1  − 1.39i·2-s + (−0.694 + 0.719i)3-s − 0.945·4-s + 1.53i·5-s + (1.00 + 0.968i)6-s + 0.265·7-s − 0.0763i·8-s + (−0.0362 − 0.999i)9-s + 2.13·10-s + 0.0561i·11-s + (0.656 − 0.680i)12-s + 0.602·13-s − 0.369i·14-s + (−1.10 − 1.06i)15-s − 1.05·16-s + 0.802i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 - 0.694i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.719 - 0.694i$
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: \((2,\ 177,\ (\ :2),\ -0.719 - 0.694i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.1590377424\)
\(L(\frac12)\) \(\approx\) \(0.1590377424\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (6.24 - 6.47i)T \)
59 \( 1 - 453. iT \)
good2 \( 1 + 5.57iT - 16T^{2} \)
5 \( 1 - 38.3iT - 625T^{2} \)
7 \( 1 - 12.9T + 2.40e3T^{2} \)
11 \( 1 - 6.78iT - 1.46e4T^{2} \)
13 \( 1 - 101.T + 2.85e4T^{2} \)
17 \( 1 - 232. iT - 8.35e4T^{2} \)
19 \( 1 + 576.T + 1.30e5T^{2} \)
23 \( 1 + 40.2iT - 2.79e5T^{2} \)
29 \( 1 + 784. iT - 7.07e5T^{2} \)
31 \( 1 + 1.04e3T + 9.23e5T^{2} \)
37 \( 1 + 2.43e3T + 1.87e6T^{2} \)
41 \( 1 + 2.58e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.12e3T + 3.41e6T^{2} \)
47 \( 1 - 2.14e3iT - 4.87e6T^{2} \)
53 \( 1 + 2.75e3iT - 7.89e6T^{2} \)
61 \( 1 - 1.24e3T + 1.38e7T^{2} \)
67 \( 1 + 7.31e3T + 2.01e7T^{2} \)
71 \( 1 - 4.74e3iT - 2.54e7T^{2} \)
73 \( 1 + 2.51e3T + 2.83e7T^{2} \)
79 \( 1 + 3.07e3T + 3.89e7T^{2} \)
83 \( 1 + 5.43e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.06e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.29e4T + 8.85e7T^{2} \)
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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

−12.01219870281395244748820238752, −10.99793411956264054310578236577, −10.74627692080051418808726263859, −10.03456617019811544987334810707, −8.766738986134380178429073429786, −6.94310081569555249586679372330, −5.98870802541316657415457181424, −4.19848703997464509578431133017, −3.38849862304670519378618231098, −1.98438134440253281250901883010, 0.06241690150929846071813661840, 1.63193084267317788956950941469, 4.60919026682921819499264933595, 5.34175077872862264483679046711, 6.30875715278757589157416561123, 7.37548576641694091752580014303, 8.395415706698615864714160808870, 8.969008706992976342433929197725, 10.79663595017020068816949623746, 11.86910177655895654488988289966

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