Properties

Label 2-177-59.58-c4-0-25
Degree $2$
Conductor $177$
Sign $-0.00986 + 0.999i$
Analytic cond. $18.2964$
Root an. cond. $4.27743$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.33i·2-s − 5.19·3-s + 10.5·4-s + 30.0·5-s + 12.1i·6-s − 41.7·7-s − 62.0i·8-s + 27·9-s − 70.2i·10-s − 117. i·11-s − 54.6·12-s + 118. i·13-s + 97.6i·14-s − 156.·15-s + 23.2·16-s + 263.·17-s + ⋯
L(s)  = 1  − 0.584i·2-s − 0.577·3-s + 0.657·4-s + 1.20·5-s + 0.337i·6-s − 0.851·7-s − 0.969i·8-s + 0.333·9-s − 0.702i·10-s − 0.970i·11-s − 0.379·12-s + 0.700i·13-s + 0.498i·14-s − 0.693·15-s + 0.0906·16-s + 0.913·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00986 + 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.00986 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.00986 + 0.999i$
Analytic conductor: \(18.2964\)
Root analytic conductor: \(4.27743\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :2),\ -0.00986 + 0.999i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.104707025\)
\(L(\frac12)\) \(\approx\) \(2.104707025\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 5.19T \)
59 \( 1 + (34.3 - 3.48e3i)T \)
good2 \( 1 + 2.33iT - 16T^{2} \)
5 \( 1 - 30.0T + 625T^{2} \)
7 \( 1 + 41.7T + 2.40e3T^{2} \)
11 \( 1 + 117. iT - 1.46e4T^{2} \)
13 \( 1 - 118. iT - 2.85e4T^{2} \)
17 \( 1 - 263.T + 8.35e4T^{2} \)
19 \( 1 - 373.T + 1.30e5T^{2} \)
23 \( 1 + 503. iT - 2.79e5T^{2} \)
29 \( 1 + 549.T + 7.07e5T^{2} \)
31 \( 1 + 450. iT - 9.23e5T^{2} \)
37 \( 1 + 1.43e3iT - 1.87e6T^{2} \)
41 \( 1 - 1.44e3T + 2.82e6T^{2} \)
43 \( 1 + 3.20e3iT - 3.41e6T^{2} \)
47 \( 1 - 967. iT - 4.87e6T^{2} \)
53 \( 1 - 1.66e3T + 7.89e6T^{2} \)
61 \( 1 - 1.71e3iT - 1.38e7T^{2} \)
67 \( 1 + 5.06e3iT - 2.01e7T^{2} \)
71 \( 1 - 6.15e3T + 2.54e7T^{2} \)
73 \( 1 - 4.17e3iT - 2.83e7T^{2} \)
79 \( 1 + 5.99e3T + 3.89e7T^{2} \)
83 \( 1 + 2.93e3iT - 4.74e7T^{2} \)
89 \( 1 + 3.87e3iT - 6.27e7T^{2} \)
97 \( 1 + 2.34e3iT - 8.85e7T^{2} \)
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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

−11.72845617694548920006433341247, −10.75202941129290845532505835401, −9.953359739628721100411915762993, −9.187785079630040215202973552977, −7.31804260646091393653174709496, −6.23701687808694142468759816272, −5.62681773665568418840495778779, −3.60798355629115272383810948675, −2.30836344071872843655135661700, −0.870416092807677147748738602970, 1.52336710763147166432009793764, 3.03747549077897921450493634830, 5.24586761250215963824776413652, 5.90285949394732116276477343842, 6.86419734581886923533204962578, 7.80060090773495715455382747654, 9.645926101817594221761956460082, 9.995834527709638076297608763791, 11.27097540633265056810821627288, 12.33830999172682468630970489674

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