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Properties

Label 2-177-59.58-c10-0-90
Degree $2$
Conductor $177$
Sign $-0.926 + 0.375i$
Analytic cond. $112.458$
Root an. cond. $10.6046$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 46.3i·2-s + 140.·3-s − 1.12e3·4-s + 3.85e3·5-s − 6.49e3i·6-s + 2.24e4·7-s + 4.45e3i·8-s + 1.96e4·9-s − 1.78e5i·10-s − 1.73e5i·11-s − 1.57e5·12-s − 6.74e5i·13-s − 1.03e6i·14-s + 5.41e5·15-s − 9.40e5·16-s + 2.43e6·17-s + ⋯
L(s)  = 1  − 1.44i·2-s + 0.577·3-s − 1.09·4-s + 1.23·5-s − 0.835i·6-s + 1.33·7-s + 0.135i·8-s + 0.333·9-s − 1.78i·10-s − 1.07i·11-s − 0.631·12-s − 1.81i·13-s − 1.93i·14-s + 0.712·15-s − 0.897·16-s + 1.71·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 + 0.375i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.926 + 0.375i$
Analytic conductor: \(112.458\)
Root analytic conductor: \(10.6046\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :5),\ -0.926 + 0.375i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(4.822496293\)
\(L(\frac12)\) \(\approx\) \(4.822496293\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 140.T \)
59 \( 1 + (6.62e8 - 2.68e8i)T \)
good2 \( 1 + 46.3iT - 1.02e3T^{2} \)
5 \( 1 - 3.85e3T + 9.76e6T^{2} \)
7 \( 1 - 2.24e4T + 2.82e8T^{2} \)
11 \( 1 + 1.73e5iT - 2.59e10T^{2} \)
13 \( 1 + 6.74e5iT - 1.37e11T^{2} \)
17 \( 1 - 2.43e6T + 2.01e12T^{2} \)
19 \( 1 - 1.33e6T + 6.13e12T^{2} \)
23 \( 1 + 5.62e5iT - 4.14e13T^{2} \)
29 \( 1 - 7.41e6T + 4.20e14T^{2} \)
31 \( 1 - 3.31e7iT - 8.19e14T^{2} \)
37 \( 1 - 7.74e7iT - 4.80e15T^{2} \)
41 \( 1 - 1.51e8T + 1.34e16T^{2} \)
43 \( 1 - 1.64e8iT - 2.16e16T^{2} \)
47 \( 1 + 2.27e8iT - 5.25e16T^{2} \)
53 \( 1 + 6.16e8T + 1.74e17T^{2} \)
61 \( 1 - 5.94e8iT - 7.13e17T^{2} \)
67 \( 1 - 1.29e9iT - 1.82e18T^{2} \)
71 \( 1 - 2.19e9T + 3.25e18T^{2} \)
73 \( 1 - 1.44e8iT - 4.29e18T^{2} \)
79 \( 1 + 5.52e9T + 9.46e18T^{2} \)
83 \( 1 + 2.03e9iT - 1.55e19T^{2} \)
89 \( 1 - 2.60e8iT - 3.11e19T^{2} \)
97 \( 1 + 9.73e9iT - 7.37e19T^{2} \)
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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

−10.36301006244277442170676728247, −9.817248145659986470546238684272, −8.587651434407394338667960962739, −7.75913394729587081165932052583, −5.85003231095518144096850964783, −4.96145406584605906804645500587, −3.30267573917229550806603526076, −2.73559760862472328291007842036, −1.38344108127087011667173073020, −1.01159781719830852557305039493, 1.52871091877025911732591186261, 2.19225822630365815352337406509, 4.31110575278406924815302389057, 5.18776795010090649237690336436, 6.16481504598559618831395505159, 7.32806486039644235673585965763, 7.917603744597549814513595478141, 9.207172433448627811967292708178, 9.689059061836109689608676259349, 11.22987506814562951150042128199

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