Properties

Label 2-177-177.176-c3-0-38
Degree $2$
Conductor $177$
Sign $-0.441 + 0.897i$
Analytic cond. $10.4433$
Root an. cond. $3.23161$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.02·2-s + (4.35 − 2.83i)3-s + 17.2·4-s + 15.7i·5-s + (−21.8 + 14.2i)6-s − 18.9·7-s − 46.3·8-s + (10.8 − 24.7i)9-s − 79.3i·10-s − 36.5·11-s + (75.0 − 48.9i)12-s − 4.20i·13-s + 95.2·14-s + (44.8 + 68.7i)15-s + 95.1·16-s − 97.3i·17-s + ⋯
L(s)  = 1  − 1.77·2-s + (0.837 − 0.546i)3-s + 2.15·4-s + 1.41i·5-s + (−1.48 + 0.970i)6-s − 1.02·7-s − 2.05·8-s + (0.403 − 0.915i)9-s − 2.50i·10-s − 1.00·11-s + (1.80 − 1.17i)12-s − 0.0897i·13-s + 1.81·14-s + (0.771 + 1.18i)15-s + 1.48·16-s − 1.38i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.441 + 0.897i$
Analytic conductor: \(10.4433\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ -0.441 + 0.897i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.227411 - 0.365324i\)
\(L(\frac12)\) \(\approx\) \(0.227411 - 0.365324i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.35 + 2.83i)T \)
59 \( 1 + (389. - 231. i)T \)
good2 \( 1 + 5.02T + 8T^{2} \)
5 \( 1 - 15.7iT - 125T^{2} \)
7 \( 1 + 18.9T + 343T^{2} \)
11 \( 1 + 36.5T + 1.33e3T^{2} \)
13 \( 1 + 4.20iT - 2.19e3T^{2} \)
17 \( 1 + 97.3iT - 4.91e3T^{2} \)
19 \( 1 + 44.1T + 6.85e3T^{2} \)
23 \( 1 - 190.T + 1.21e4T^{2} \)
29 \( 1 + 131. iT - 2.43e4T^{2} \)
31 \( 1 + 206. iT - 2.97e4T^{2} \)
37 \( 1 + 106. iT - 5.06e4T^{2} \)
41 \( 1 + 214. iT - 6.89e4T^{2} \)
43 \( 1 + 213. iT - 7.95e4T^{2} \)
47 \( 1 + 444.T + 1.03e5T^{2} \)
53 \( 1 - 624. iT - 1.48e5T^{2} \)
61 \( 1 + 437. iT - 2.26e5T^{2} \)
67 \( 1 - 648. iT - 3.00e5T^{2} \)
71 \( 1 + 809. iT - 3.57e5T^{2} \)
73 \( 1 - 39.3iT - 3.89e5T^{2} \)
79 \( 1 - 595.T + 4.93e5T^{2} \)
83 \( 1 + 642.T + 5.71e5T^{2} \)
89 \( 1 + 258.T + 7.04e5T^{2} \)
97 \( 1 + 1.81e3iT - 9.12e5T^{2} \)
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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

−11.47423073398661350548271784136, −10.56279460866576672751510188273, −9.742252684023914629754873395228, −8.995136869555538527542511163654, −7.68540805174500131424552254438, −7.15378388783656422887728677912, −6.32680222433094734209950813543, −3.05339737355306789075037170358, −2.44560635793978728481055317178, −0.30774461172463386428779746569, 1.47063273166797278744096263859, 3.09317577375831927534316182230, 4.95796031686127627593432521999, 6.70509760253972808637778973650, 8.084974427577663006612473645737, 8.601445652302177716696051776928, 9.367046955644583029244122635996, 10.15351100539682428494246759491, 10.98174439966234650341714921492, 12.74022583779035667289077436839

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