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Properties

Label 2-177-59.58-c10-0-36
Degree $2$
Conductor $177$
Sign $-0.404 + 0.914i$
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  + 59.5i·2-s + 140.·3-s − 2.52e3·4-s − 4.76e3·5-s + 8.35e3i·6-s + 8.81e3·7-s − 8.94e4i·8-s + 1.96e4·9-s − 2.83e5i·10-s + 3.07e5i·11-s − 3.54e5·12-s + 6.89e4i·13-s + 5.25e5i·14-s − 6.68e5·15-s + 2.74e6·16-s + 1.05e6·17-s + ⋯
L(s)  = 1  + 1.86i·2-s + 0.577·3-s − 2.46·4-s − 1.52·5-s + 1.07i·6-s + 0.524·7-s − 2.72i·8-s + 0.333·9-s − 2.83i·10-s + 1.90i·11-s − 1.42·12-s + 0.185i·13-s + 0.976i·14-s − 0.880·15-s + 2.61·16-s + 0.744·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.404 + 0.914i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.404 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.404 + 0.914i$
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: \((2,\ 177,\ (\ :5),\ -0.404 + 0.914i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.744359770\)
\(L(\frac12)\) \(\approx\) \(1.744359770\)
\(L(6)\) 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 \( 1 + (2.89e8 - 6.53e8i)T \)
good2 \( 1 - 59.5iT - 1.02e3T^{2} \)
5 \( 1 + 4.76e3T + 9.76e6T^{2} \)
7 \( 1 - 8.81e3T + 2.82e8T^{2} \)
11 \( 1 - 3.07e5iT - 2.59e10T^{2} \)
13 \( 1 - 6.89e4iT - 1.37e11T^{2} \)
17 \( 1 - 1.05e6T + 2.01e12T^{2} \)
19 \( 1 - 4.26e6T + 6.13e12T^{2} \)
23 \( 1 - 3.74e6iT - 4.14e13T^{2} \)
29 \( 1 - 1.29e7T + 4.20e14T^{2} \)
31 \( 1 - 5.38e7iT - 8.19e14T^{2} \)
37 \( 1 - 2.46e7iT - 4.80e15T^{2} \)
41 \( 1 - 1.00e8T + 1.34e16T^{2} \)
43 \( 1 - 1.57e8iT - 2.16e16T^{2} \)
47 \( 1 + 1.16e8iT - 5.25e16T^{2} \)
53 \( 1 - 6.16e8T + 1.74e17T^{2} \)
61 \( 1 + 9.65e7iT - 7.13e17T^{2} \)
67 \( 1 + 1.44e9iT - 1.82e18T^{2} \)
71 \( 1 - 1.91e9T + 3.25e18T^{2} \)
73 \( 1 + 1.00e9iT - 4.29e18T^{2} \)
79 \( 1 - 4.68e8T + 9.46e18T^{2} \)
83 \( 1 + 3.18e9iT - 1.55e19T^{2} \)
89 \( 1 - 7.27e9iT - 3.11e19T^{2} \)
97 \( 1 + 1.55e10iT - 7.37e19T^{2} \)
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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

−11.83412491867454134248235375198, −10.01792710776889950920674439144, −9.075210530079944939523348805168, −7.959992954310952524758974526673, −7.54053299199686030329570587678, −6.89561799729121770452223925753, −5.11540207505765921067053899602, −4.49682745386125234983609410963, −3.41498635244729355297144109246, −1.13807648272786902773801706019, 0.50668984447706318465408428214, 0.978649611489196546772780869316, 2.66548587357407100921858327960, 3.45021379384958828792162290270, 4.08768063598526671318736911159, 5.40136375007269240295411071504, 7.77265349966406944985212732544, 8.306914480098230145846773754947, 9.264731155760180353700648200054, 10.46575927042054042977698327764

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