Properties

Label 2-177-177.56-c1-0-16
Degree $2$
Conductor $177$
Sign $0.551 + 0.833i$
Analytic cond. $1.41335$
Root an. cond. $1.18884$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.81 − 0.841i)2-s + (1.13 − 1.30i)3-s + (1.30 − 1.53i)4-s + (−1.80 + 2.99i)5-s + (0.970 − 3.33i)6-s + (0.204 − 3.77i)7-s + (0.00711 − 0.0256i)8-s + (−0.411 − 2.97i)9-s + (−0.756 + 6.95i)10-s + (−0.661 + 4.03i)11-s + (−0.521 − 3.44i)12-s + (−0.0782 − 0.102i)13-s + (−2.80 − 7.03i)14-s + (1.85 + 5.75i)15-s + (0.642 + 3.92i)16-s + (2.21 − 0.120i)17-s + ⋯
L(s)  = 1  + (1.28 − 0.594i)2-s + (0.656 − 0.754i)3-s + (0.651 − 0.767i)4-s + (−0.805 + 1.33i)5-s + (0.396 − 1.36i)6-s + (0.0773 − 1.42i)7-s + (0.00251 − 0.00906i)8-s + (−0.137 − 0.990i)9-s + (−0.239 + 2.19i)10-s + (−0.199 + 1.21i)11-s + (−0.150 − 0.995i)12-s + (−0.0217 − 0.0285i)13-s + (−0.749 − 1.88i)14-s + (0.480 + 1.48i)15-s + (0.160 + 0.980i)16-s + (0.537 − 0.0291i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.551 + 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.551 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.551 + 0.833i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 0.551 + 0.833i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96379 - 1.05529i\)
\(L(\frac12)\) \(\approx\) \(1.96379 - 1.05529i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.13 + 1.30i)T \)
59 \( 1 + (3.04 - 7.05i)T \)
good2 \( 1 + (-1.81 + 0.841i)T + (1.29 - 1.52i)T^{2} \)
5 \( 1 + (1.80 - 2.99i)T + (-2.34 - 4.41i)T^{2} \)
7 \( 1 + (-0.204 + 3.77i)T + (-6.95 - 0.756i)T^{2} \)
11 \( 1 + (0.661 - 4.03i)T + (-10.4 - 3.51i)T^{2} \)
13 \( 1 + (0.0782 + 0.102i)T + (-3.47 + 12.5i)T^{2} \)
17 \( 1 + (-2.21 + 0.120i)T + (16.9 - 1.83i)T^{2} \)
19 \( 1 + (1.70 - 1.61i)T + (1.02 - 18.9i)T^{2} \)
23 \( 1 + (5.79 - 1.27i)T + (20.8 - 9.65i)T^{2} \)
29 \( 1 + (-3.33 + 7.20i)T + (-18.7 - 22.1i)T^{2} \)
31 \( 1 + (2.36 - 2.49i)T + (-1.67 - 30.9i)T^{2} \)
37 \( 1 + (-3.94 + 1.09i)T + (31.7 - 19.0i)T^{2} \)
41 \( 1 + (-1.94 + 8.83i)T + (-37.2 - 17.2i)T^{2} \)
43 \( 1 + (-7.68 + 1.26i)T + (40.7 - 13.7i)T^{2} \)
47 \( 1 + (5.56 - 3.34i)T + (22.0 - 41.5i)T^{2} \)
53 \( 1 + (0.648 + 5.96i)T + (-51.7 + 11.3i)T^{2} \)
61 \( 1 + (1.61 + 3.49i)T + (-39.4 + 46.4i)T^{2} \)
67 \( 1 + (-4.01 - 1.11i)T + (57.4 + 34.5i)T^{2} \)
71 \( 1 + (2.69 + 4.48i)T + (-33.2 + 62.7i)T^{2} \)
73 \( 1 + (0.805 - 0.321i)T + (52.9 - 50.2i)T^{2} \)
79 \( 1 + (-14.0 + 4.74i)T + (62.8 - 47.8i)T^{2} \)
83 \( 1 + (2.40 - 3.55i)T + (-30.7 - 77.1i)T^{2} \)
89 \( 1 + (10.7 + 4.97i)T + (57.6 + 67.8i)T^{2} \)
97 \( 1 + (-9.59 - 3.82i)T + (70.4 + 66.7i)T^{2} \)
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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

−12.56100404420236196234057828142, −11.86949505520914032584875871814, −10.82538938642318398527935006306, −9.991235100449737224188404031584, −7.898049247175682726501405444580, −7.35550272200225407268311855755, −6.28005110572079409503684446143, −4.23536164766032815358573343208, −3.57775785029908335374932893871, −2.26677622330349626575101016518, 3.01978335120005503969588560011, 4.23921764680181921094893396331, 5.14456010796272537678525039977, 5.94927027212029585264499156902, 7.965731357456206222268911856739, 8.598596227631484969842551224480, 9.511097686079748654977970072469, 11.25261531111389305763577909509, 12.26117750526575462524919687786, 12.88970291024771798023918980288

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