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Properties

Label 2-177-177.176-c3-0-34
Degree $2$
Conductor $177$
Sign $-0.379 + 0.925i$
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  − 2.68·2-s + (−3.19 − 4.09i)3-s − 0.797·4-s − 15.8i·5-s + (8.57 + 10.9i)6-s + 25.4·7-s + 23.6·8-s + (−6.56 + 26.1i)9-s + 42.4i·10-s + 64.0·11-s + (2.54 + 3.26i)12-s − 26.7i·13-s − 68.2·14-s + (−64.7 + 50.5i)15-s − 56.9·16-s − 54.3i·17-s + ⋯
L(s)  = 1  − 0.948·2-s + (−0.615 − 0.788i)3-s − 0.0997·4-s − 1.41i·5-s + (0.583 + 0.748i)6-s + 1.37·7-s + 1.04·8-s + (−0.243 + 0.969i)9-s + 1.34i·10-s + 1.75·11-s + (0.0613 + 0.0786i)12-s − 0.570i·13-s − 1.30·14-s + (−1.11 + 0.870i)15-s − 0.890·16-s − 0.775i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.379 + 0.925i$
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: \((2,\ 177,\ (\ :3/2),\ -0.379 + 0.925i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.553007 - 0.824152i\)
\(L(\frac12)\) \(\approx\) \(0.553007 - 0.824152i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.19 + 4.09i)T \)
59 \( 1 + (224. + 393. i)T \)
good2 \( 1 + 2.68T + 8T^{2} \)
5 \( 1 + 15.8iT - 125T^{2} \)
7 \( 1 - 25.4T + 343T^{2} \)
11 \( 1 - 64.0T + 1.33e3T^{2} \)
13 \( 1 + 26.7iT - 2.19e3T^{2} \)
17 \( 1 + 54.3iT - 4.91e3T^{2} \)
19 \( 1 - 47.4T + 6.85e3T^{2} \)
23 \( 1 - 174.T + 1.21e4T^{2} \)
29 \( 1 + 169. iT - 2.43e4T^{2} \)
31 \( 1 - 217. iT - 2.97e4T^{2} \)
37 \( 1 + 122. iT - 5.06e4T^{2} \)
41 \( 1 - 368. iT - 6.89e4T^{2} \)
43 \( 1 - 3.01iT - 7.95e4T^{2} \)
47 \( 1 + 314.T + 1.03e5T^{2} \)
53 \( 1 + 468. iT - 1.48e5T^{2} \)
61 \( 1 - 558. iT - 2.26e5T^{2} \)
67 \( 1 - 284. iT - 3.00e5T^{2} \)
71 \( 1 - 279. iT - 3.57e5T^{2} \)
73 \( 1 - 877. iT - 3.89e5T^{2} \)
79 \( 1 + 63.0T + 4.93e5T^{2} \)
83 \( 1 - 941.T + 5.71e5T^{2} \)
89 \( 1 + 199.T + 7.04e5T^{2} \)
97 \( 1 + 432. iT - 9.12e5T^{2} \)
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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

−11.73015360720744894087590131911, −11.16946808069488580343696995255, −9.628994796468105626912777041807, −8.712380803160626519035091123653, −8.076020738579953270896537087284, −6.98899084924120852012300792816, −5.25052671533165553212238628035, −4.59770877983175183703174968110, −1.40962850918489519539349617188, −0.914060490225708671213291151452, 1.40190158591476565488047956375, 3.71629282806275236748595991793, 4.83160369748694343393772073424, 6.43103679271381678239845710426, 7.39461246665669806985459732254, 8.801365763035485472525553185754, 9.504568354903394895001251819936, 10.74787340756897929129883020109, 11.06481168241411059381957822857, 11.95468441273959188314870702056

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