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Properties

Label 2-177-59.58-c10-0-37
Degree $2$
Conductor $177$
Sign $0.509 + 0.860i$
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  − 59.8i·2-s + 140.·3-s − 2.56e3·4-s + 3.68e3·5-s − 8.40e3i·6-s + 4.18e3·7-s + 9.21e4i·8-s + 1.96e4·9-s − 2.20e5i·10-s − 1.08e5i·11-s − 3.59e5·12-s + 5.82e5i·13-s − 2.50e5i·14-s + 5.17e5·15-s + 2.89e6·16-s + 7.33e5·17-s + ⋯
L(s)  = 1  − 1.87i·2-s + 0.577·3-s − 2.50·4-s + 1.18·5-s − 1.08i·6-s + 0.249·7-s + 2.81i·8-s + 0.333·9-s − 2.20i·10-s − 0.674i·11-s − 1.44·12-s + 1.57i·13-s − 0.466i·14-s + 0.681·15-s + 2.75·16-s + 0.516·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.509 + 0.860i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.509 + 0.860i$
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: \((2,\ 177,\ (\ :5),\ 0.509 + 0.860i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(2.581639883\)
\(L(\frac12)\) \(\approx\) \(2.581639883\)
\(L(6)\) 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 \( 1 + (-3.64e8 - 6.15e8i)T \)
good2 \( 1 + 59.8iT - 1.02e3T^{2} \)
5 \( 1 - 3.68e3T + 9.76e6T^{2} \)
7 \( 1 - 4.18e3T + 2.82e8T^{2} \)
11 \( 1 + 1.08e5iT - 2.59e10T^{2} \)
13 \( 1 - 5.82e5iT - 1.37e11T^{2} \)
17 \( 1 - 7.33e5T + 2.01e12T^{2} \)
19 \( 1 + 2.47e6T + 6.13e12T^{2} \)
23 \( 1 - 2.73e5iT - 4.14e13T^{2} \)
29 \( 1 + 7.13e6T + 4.20e14T^{2} \)
31 \( 1 - 8.58e6iT - 8.19e14T^{2} \)
37 \( 1 - 1.12e8iT - 4.80e15T^{2} \)
41 \( 1 - 5.07e7T + 1.34e16T^{2} \)
43 \( 1 - 6.60e7iT - 2.16e16T^{2} \)
47 \( 1 - 4.56e8iT - 5.25e16T^{2} \)
53 \( 1 - 5.34e8T + 1.74e17T^{2} \)
61 \( 1 + 5.64e8iT - 7.13e17T^{2} \)
67 \( 1 + 3.92e8iT - 1.82e18T^{2} \)
71 \( 1 - 2.57e9T + 3.25e18T^{2} \)
73 \( 1 + 1.12e9iT - 4.29e18T^{2} \)
79 \( 1 - 2.65e9T + 9.46e18T^{2} \)
83 \( 1 + 1.54e8iT - 1.55e19T^{2} \)
89 \( 1 - 7.28e9iT - 3.11e19T^{2} \)
97 \( 1 - 3.03e9iT - 7.37e19T^{2} \)
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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

−10.69022524090650024161925248854, −9.709534078199077996703554456153, −9.176444557080898010337382393697, −8.239978834018157194347064435262, −6.31445987812280253050241830853, −4.90686661837531292985817591369, −3.88418378353354525145477779803, −2.71012807390622631261318835741, −1.88554688669686773273677291001, −1.15300065394888374891275444887, 0.51849941890355244240862434182, 2.14757819675461344877080264225, 3.85884463819645215899707324263, 5.19891259403158590290949762115, 5.80671234669970525376732547852, 6.94308324019877138542997285121, 7.85757568547938111853700893969, 8.683918695356436382870334302426, 9.664004862513362569133793627457, 10.37377006558405729027948363142

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