Properties

Label 2-177-177.104-c2-0-23
Degree $2$
Conductor $177$
Sign $0.732 + 0.680i$
Analytic cond. $4.82290$
Root an. cond. $2.19611$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.328 + 0.546i)2-s + (0.396 − 2.97i)3-s + (1.68 − 3.17i)4-s + (0.662 + 6.09i)5-s + (1.75 − 0.760i)6-s + (7.88 − 2.65i)7-s + (4.83 − 0.262i)8-s + (−8.68 − 2.35i)9-s + (−3.11 + 2.36i)10-s + (2.49 + 1.69i)11-s + (−8.77 − 6.26i)12-s + (−0.610 − 0.578i)13-s + (4.04 + 3.43i)14-s + (18.3 + 0.444i)15-s + (−6.33 − 9.34i)16-s + (5.17 − 15.3i)17-s + ⋯
L(s)  = 1  + (0.164 + 0.273i)2-s + (0.132 − 0.991i)3-s + (0.420 − 0.793i)4-s + (0.132 + 1.21i)5-s + (0.292 − 0.126i)6-s + (1.12 − 0.379i)7-s + (0.604 − 0.0327i)8-s + (−0.965 − 0.261i)9-s + (−0.311 + 0.236i)10-s + (0.226 + 0.153i)11-s + (−0.731 − 0.521i)12-s + (−0.0469 − 0.0444i)13-s + (0.288 + 0.245i)14-s + (1.22 + 0.0296i)15-s + (−0.395 − 0.583i)16-s + (0.304 − 0.902i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 + 0.680i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.732 + 0.680i$
Analytic conductor: \(4.82290\)
Root analytic conductor: \(2.19611\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1),\ 0.732 + 0.680i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.90217 - 0.746995i\)
\(L(\frac12)\) \(\approx\) \(1.90217 - 0.746995i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.396 + 2.97i)T \)
59 \( 1 + (56.5 + 16.8i)T \)
good2 \( 1 + (-0.328 - 0.546i)T + (-1.87 + 3.53i)T^{2} \)
5 \( 1 + (-0.662 - 6.09i)T + (-24.4 + 5.37i)T^{2} \)
7 \( 1 + (-7.88 + 2.65i)T + (39.0 - 29.6i)T^{2} \)
11 \( 1 + (-2.49 - 1.69i)T + (44.7 + 112. i)T^{2} \)
13 \( 1 + (0.610 + 0.578i)T + (9.14 + 168. i)T^{2} \)
17 \( 1 + (-5.17 + 15.3i)T + (-230. - 174. i)T^{2} \)
19 \( 1 + (1.70 + 10.3i)T + (-342. + 115. i)T^{2} \)
23 \( 1 + (4.13 - 1.14i)T + (453. - 272. i)T^{2} \)
29 \( 1 + (9.80 - 16.3i)T + (-393. - 743. i)T^{2} \)
31 \( 1 + (7.81 - 47.6i)T + (-910. - 306. i)T^{2} \)
37 \( 1 + (-0.869 + 16.0i)T + (-1.36e3 - 148. i)T^{2} \)
41 \( 1 + (-13.3 - 3.69i)T + (1.44e3 + 866. i)T^{2} \)
43 \( 1 + (-28.5 - 42.0i)T + (-684. + 1.71e3i)T^{2} \)
47 \( 1 + (1.68 - 15.4i)T + (-2.15e3 - 474. i)T^{2} \)
53 \( 1 + (62.5 - 82.3i)T + (-751. - 2.70e3i)T^{2} \)
61 \( 1 + (25.7 - 15.5i)T + (1.74e3 - 3.28e3i)T^{2} \)
67 \( 1 + (3.13 + 57.8i)T + (-4.46e3 + 485. i)T^{2} \)
71 \( 1 + (10.1 - 92.8i)T + (-4.92e3 - 1.08e3i)T^{2} \)
73 \( 1 + (-89.9 + 105. i)T + (-862. - 5.25e3i)T^{2} \)
79 \( 1 + (14.5 - 36.5i)T + (-4.53e3 - 4.29e3i)T^{2} \)
83 \( 1 + (-31.4 - 67.9i)T + (-4.45e3 + 5.25e3i)T^{2} \)
89 \( 1 + (-31.4 + 52.2i)T + (-3.71e3 - 6.99e3i)T^{2} \)
97 \( 1 + (65.1 + 76.7i)T + (-1.52e3 + 9.28e3i)T^{2} \)
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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

−12.27462622925804745958128236055, −11.09210079168097860216913593552, −10.82280910141940548361537854964, −9.323150886217032928537761260285, −7.72470805492146798043886074406, −7.10992050520841198532279640328, −6.23519286049923755602943606244, −4.96338877603759628780628826149, −2.80004876454090469290885404346, −1.43161388775351034701505533455, 1.98009856325964903057529121418, 3.79767074533861937550985806788, 4.70144480430321272667915754452, 5.82441836270954018243409018890, 7.950671033733926684296352390057, 8.431789121904701599069074743575, 9.481557595224644474348445794027, 10.78973008237078810667579571716, 11.62643291161508423679554425344, 12.38816508670252107972341651628

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