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Properties

Label 2-177-3.2-c4-0-10
Degree $2$
Conductor $177$
Sign $-0.456 + 0.889i$
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  + 5.88i·2-s + (−8.00 − 4.11i)3-s − 18.6·4-s + 15.2i·5-s + (24.1 − 47.0i)6-s + 49.3·7-s − 15.3i·8-s + (47.1 + 65.8i)9-s − 89.6·10-s + 123. i·11-s + (148. + 76.5i)12-s − 224.·13-s + 290. i·14-s + (62.6 − 122. i)15-s − 207.·16-s + 169. i·17-s + ⋯
L(s)  = 1  + 1.47i·2-s + (−0.889 − 0.456i)3-s − 1.16·4-s + 0.609i·5-s + (0.672 − 1.30i)6-s + 1.00·7-s − 0.239i·8-s + (0.582 + 0.812i)9-s − 0.896·10-s + 1.02i·11-s + (1.03 + 0.531i)12-s − 1.33·13-s + 1.48i·14-s + (0.278 − 0.542i)15-s − 0.810·16-s + 0.586i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.456 + 0.889i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.456 + 0.889i$
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: \((2,\ 177,\ (\ :2),\ -0.456 + 0.889i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.7381129586\)
\(L(\frac12)\) \(\approx\) \(0.7381129586\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (8.00 + 4.11i)T \)
59 \( 1 + 453. iT \)
good2 \( 1 - 5.88iT - 16T^{2} \)
5 \( 1 - 15.2iT - 625T^{2} \)
7 \( 1 - 49.3T + 2.40e3T^{2} \)
11 \( 1 - 123. iT - 1.46e4T^{2} \)
13 \( 1 + 224.T + 2.85e4T^{2} \)
17 \( 1 - 169. iT - 8.35e4T^{2} \)
19 \( 1 - 279.T + 1.30e5T^{2} \)
23 \( 1 - 54.0iT - 2.79e5T^{2} \)
29 \( 1 + 809. iT - 7.07e5T^{2} \)
31 \( 1 + 1.47e3T + 9.23e5T^{2} \)
37 \( 1 + 139.T + 1.87e6T^{2} \)
41 \( 1 + 769. iT - 2.82e6T^{2} \)
43 \( 1 + 2.76e3T + 3.41e6T^{2} \)
47 \( 1 - 3.56e3iT - 4.87e6T^{2} \)
53 \( 1 + 2.80e3iT - 7.89e6T^{2} \)
61 \( 1 + 5.10e3T + 1.38e7T^{2} \)
67 \( 1 + 2.15e3T + 2.01e7T^{2} \)
71 \( 1 + 2.28e3iT - 2.54e7T^{2} \)
73 \( 1 - 7.25e3T + 2.83e7T^{2} \)
79 \( 1 - 1.08e3T + 3.89e7T^{2} \)
83 \( 1 - 1.10e3iT - 4.74e7T^{2} \)
89 \( 1 - 9.27e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.69e4T + 8.85e7T^{2} \)
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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

−12.70693568933745384460417617645, −11.72953071542038403213517882777, −10.78269145649131702037898482457, −9.574050720057566801094727833393, −7.932612828837333990679308300997, −7.38635040211617959635500719619, −6.58779867467718538415963395013, −5.34613001487126668130337592343, −4.66144783806201287395571413980, −1.98099017258773995062579283682, 0.31193449696427664599217554752, 1.50058478338839361410533034625, 3.25435151983529762933429974892, 4.68605210840646281831229706830, 5.31407415267973747350438512095, 7.15225990862618640807120522558, 8.763158009412649776774235986943, 9.655954506408965128191680531415, 10.65509818586348223509326438000, 11.38254927829770979140947399115

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