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Properties

Label 2-177-59.58-c4-0-2
Degree $2$
Conductor $177$
Sign $-0.444 + 0.895i$
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  + 5.77i·2-s − 5.19·3-s − 17.3·4-s + 34.3·5-s − 29.9i·6-s − 39.2·7-s − 7.56i·8-s + 27·9-s + 198. i·10-s − 56.8i·11-s + 89.9·12-s + 232. i·13-s − 226. i·14-s − 178.·15-s − 233.·16-s − 391.·17-s + ⋯
L(s)  = 1  + 1.44i·2-s − 0.577·3-s − 1.08·4-s + 1.37·5-s − 0.833i·6-s − 0.801·7-s − 0.118i·8-s + 0.333·9-s + 1.98i·10-s − 0.469i·11-s + 0.624·12-s + 1.37i·13-s − 1.15i·14-s − 0.793·15-s − 0.911·16-s − 1.35·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.444 + 0.895i$
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: \((2,\ 177,\ (\ :2),\ -0.444 + 0.895i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.7233871187\)
\(L(\frac12)\) \(\approx\) \(0.7233871187\)
\(L(3)\) 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 \( 1 + (1.54e3 - 3.11e3i)T \)
good2 \( 1 - 5.77iT - 16T^{2} \)
5 \( 1 - 34.3T + 625T^{2} \)
7 \( 1 + 39.2T + 2.40e3T^{2} \)
11 \( 1 + 56.8iT - 1.46e4T^{2} \)
13 \( 1 - 232. iT - 2.85e4T^{2} \)
17 \( 1 + 391.T + 8.35e4T^{2} \)
19 \( 1 + 352.T + 1.30e5T^{2} \)
23 \( 1 - 173. iT - 2.79e5T^{2} \)
29 \( 1 - 90.7T + 7.07e5T^{2} \)
31 \( 1 + 128. iT - 9.23e5T^{2} \)
37 \( 1 - 413. iT - 1.87e6T^{2} \)
41 \( 1 + 761.T + 2.82e6T^{2} \)
43 \( 1 + 874. iT - 3.41e6T^{2} \)
47 \( 1 + 895. iT - 4.87e6T^{2} \)
53 \( 1 - 3.68e3T + 7.89e6T^{2} \)
61 \( 1 + 1.49e3iT - 1.38e7T^{2} \)
67 \( 1 - 3.68e3iT - 2.01e7T^{2} \)
71 \( 1 + 9.23e3T + 2.54e7T^{2} \)
73 \( 1 + 4.30e3iT - 2.83e7T^{2} \)
79 \( 1 + 4.70e3T + 3.89e7T^{2} \)
83 \( 1 - 398. iT - 4.74e7T^{2} \)
89 \( 1 - 1.24e4iT - 6.27e7T^{2} \)
97 \( 1 + 397. iT - 8.85e7T^{2} \)
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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

−13.25189842873369345363212690616, −11.69629186033938691394061501288, −10.52085902288214753384404352255, −9.361916374556345192711423869379, −8.695613876017793615280775255717, −6.93531734894113324355575057209, −6.42593780388775549528789396330, −5.69433877806528254175504649559, −4.43813580620703121690637557348, −2.10639894315086816830803794966, 0.26384579991923362018719698794, 1.83589294007216688322623115807, 2.89555870759909995501286262742, 4.49844984105966748980079289940, 5.88155886486416432521079919180, 6.80601899447201424379164640093, 8.829095356087646481737814308723, 9.865658976802396617283012326493, 10.33849020163951389465504570169, 11.15168578423764386268585431099

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