Properties

Label 2-177-3.2-c4-0-18
Degree $2$
Conductor $177$
Sign $-0.0161 - 0.999i$
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  − 1.22i·2-s + (−8.99 + 0.144i)3-s + 14.5·4-s + 37.1i·5-s + (0.177 + 11.0i)6-s + 37.9·7-s − 37.3i·8-s + (80.9 − 2.60i)9-s + 45.3·10-s + 58.5i·11-s + (−130. + 2.10i)12-s − 183.·13-s − 46.4i·14-s + (−5.38 − 334. i)15-s + 186.·16-s + 2.65i·17-s + ⋯
L(s)  = 1  − 0.305i·2-s + (−0.999 + 0.0161i)3-s + 0.906·4-s + 1.48i·5-s + (0.00492 + 0.305i)6-s + 0.774·7-s − 0.582i·8-s + (0.999 − 0.0322i)9-s + 0.453·10-s + 0.483i·11-s + (−0.906 + 0.0145i)12-s − 1.08·13-s − 0.236i·14-s + (−0.0239 − 1.48i)15-s + 0.728·16-s + 0.00920i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0161 - 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0161 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.0161 - 0.999i$
Analytic conductor: \(18.2964\)
Root analytic conductor: \(4.27743\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :2),\ -0.0161 - 0.999i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.473647044\)
\(L(\frac12)\) \(\approx\) \(1.473647044\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (8.99 - 0.144i)T \)
59 \( 1 - 453. iT \)
good2 \( 1 + 1.22iT - 16T^{2} \)
5 \( 1 - 37.1iT - 625T^{2} \)
7 \( 1 - 37.9T + 2.40e3T^{2} \)
11 \( 1 - 58.5iT - 1.46e4T^{2} \)
13 \( 1 + 183.T + 2.85e4T^{2} \)
17 \( 1 - 2.65iT - 8.35e4T^{2} \)
19 \( 1 + 170.T + 1.30e5T^{2} \)
23 \( 1 - 324. iT - 2.79e5T^{2} \)
29 \( 1 - 1.20e3iT - 7.07e5T^{2} \)
31 \( 1 - 837.T + 9.23e5T^{2} \)
37 \( 1 - 702.T + 1.87e6T^{2} \)
41 \( 1 - 2.25e3iT - 2.82e6T^{2} \)
43 \( 1 + 1.09e3T + 3.41e6T^{2} \)
47 \( 1 - 1.87e3iT - 4.87e6T^{2} \)
53 \( 1 - 601. iT - 7.89e6T^{2} \)
61 \( 1 + 4.01e3T + 1.38e7T^{2} \)
67 \( 1 - 905.T + 2.01e7T^{2} \)
71 \( 1 - 7.04e3iT - 2.54e7T^{2} \)
73 \( 1 + 7.26e3T + 2.83e7T^{2} \)
79 \( 1 - 2.27e3T + 3.89e7T^{2} \)
83 \( 1 + 8.81e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.14e4iT - 6.27e7T^{2} \)
97 \( 1 - 9.25e3T + 8.85e7T^{2} \)
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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

−11.86054808338674632078733863500, −11.29112145324396002996124504156, −10.51893747029141133237504346925, −9.862724878615283955342651255834, −7.63829138138932928781806341942, −7.00056618918617850819790171509, −6.10981492521194977436672585329, −4.67625819193293484927874229522, −2.96960316414796987753926251703, −1.66615692185296094507060518511, 0.60226205330381560996204954483, 1.98268063724393602341575862628, 4.49226845332902483959870606866, 5.28845143529733806479156915298, 6.30573088154207385627885933769, 7.59399999633182885489672191406, 8.472328985460727254892835057142, 9.895235590100581421490225281036, 10.97917357923937943451416782049, 11.92831077100892523230733507453

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