Properties

Label 2-177-177.101-c1-0-1
Degree $2$
Conductor $177$
Sign $-0.801 - 0.598i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.154 + 0.227i)2-s + (−0.819 + 1.52i)3-s + (0.712 + 1.78i)4-s + (−1.09 + 2.37i)5-s + (−0.221 − 0.422i)6-s + (−0.787 − 2.83i)7-s + (−1.05 − 0.232i)8-s + (−1.65 − 2.50i)9-s + (−0.370 − 0.616i)10-s + (−1.07 − 1.02i)11-s + (−3.31 − 0.378i)12-s + (−0.324 + 2.98i)13-s + (0.767 + 0.258i)14-s + (−2.71 − 3.61i)15-s + (−2.57 + 2.44i)16-s + (6.60 + 1.83i)17-s + ⋯
L(s)  = 1  + (−0.109 + 0.161i)2-s + (−0.473 + 0.880i)3-s + (0.356 + 0.893i)4-s + (−0.490 + 1.06i)5-s + (−0.0902 − 0.172i)6-s + (−0.297 − 1.07i)7-s + (−0.372 − 0.0820i)8-s + (−0.552 − 0.833i)9-s + (−0.117 − 0.194i)10-s + (−0.325 − 0.308i)11-s + (−0.955 − 0.109i)12-s + (−0.0899 + 0.826i)13-s + (0.205 + 0.0691i)14-s + (−0.702 − 0.934i)15-s + (−0.644 + 0.610i)16-s + (1.60 + 0.444i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.801 - 0.598i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ -0.801 - 0.598i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.256317 + 0.771378i\)
\(L(\frac12)\) \(\approx\) \(0.256317 + 0.771378i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.819 - 1.52i)T \)
59 \( 1 + (-6.44 + 4.17i)T \)
good2 \( 1 + (0.154 - 0.227i)T + (-0.740 - 1.85i)T^{2} \)
5 \( 1 + (1.09 - 2.37i)T + (-3.23 - 3.81i)T^{2} \)
7 \( 1 + (0.787 + 2.83i)T + (-5.99 + 3.60i)T^{2} \)
11 \( 1 + (1.07 + 1.02i)T + (0.595 + 10.9i)T^{2} \)
13 \( 1 + (0.324 - 2.98i)T + (-12.6 - 2.79i)T^{2} \)
17 \( 1 + (-6.60 - 1.83i)T + (14.5 + 8.76i)T^{2} \)
19 \( 1 + (-1.91 - 1.45i)T + (5.08 + 18.3i)T^{2} \)
23 \( 1 + (-3.50 - 6.61i)T + (-12.9 + 19.0i)T^{2} \)
29 \( 1 + (6.12 - 4.15i)T + (10.7 - 26.9i)T^{2} \)
31 \( 1 + (4.49 + 5.91i)T + (-8.29 + 29.8i)T^{2} \)
37 \( 1 + (0.431 + 1.96i)T + (-33.5 + 15.5i)T^{2} \)
41 \( 1 + (2.48 + 1.31i)T + (23.0 + 33.9i)T^{2} \)
43 \( 1 + (-4.78 - 5.05i)T + (-2.32 + 42.9i)T^{2} \)
47 \( 1 + (-7.93 + 3.67i)T + (30.4 - 35.8i)T^{2} \)
53 \( 1 + (1.20 - 2.00i)T + (-24.8 - 46.8i)T^{2} \)
61 \( 1 + (-10.2 - 6.98i)T + (22.5 + 56.6i)T^{2} \)
67 \( 1 + (-1.31 + 5.95i)T + (-60.8 - 28.1i)T^{2} \)
71 \( 1 + (0.638 + 1.37i)T + (-45.9 + 54.1i)T^{2} \)
73 \( 1 + (0.995 - 2.95i)T + (-58.1 - 44.1i)T^{2} \)
79 \( 1 + (-0.685 + 12.6i)T + (-78.5 - 8.54i)T^{2} \)
83 \( 1 + (-1.16 + 7.10i)T + (-78.6 - 26.5i)T^{2} \)
89 \( 1 + (5.81 + 8.57i)T + (-32.9 + 82.6i)T^{2} \)
97 \( 1 + (-0.296 - 0.879i)T + (-77.2 + 58.7i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.98364325599633501394207490453, −11.73697928217625901470093115740, −11.14843009592630128931587452506, −10.29242975266089525036535927153, −9.209251482155195652511271758901, −7.60477181089819166180963117910, −7.10937662708509962157391963027, −5.71787245026561326679284581636, −3.80731487675416615590681325163, −3.37553745014991375201966291925, 0.840525129204220449961659375973, 2.58919159521125800567061840646, 5.22047497449214034170508393691, 5.58838624788083738297407931836, 7.04631757210011198463978674563, 8.202420482987091163192699764484, 9.236879688105339053595407070932, 10.42123898264496532964590208983, 11.55440798607293227657508262430, 12.38661597777260028137795897983

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