Properties

Label 2-177-177.89-c1-0-2
Degree $2$
Conductor $177$
Sign $-0.888 - 0.459i$
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  + (0.0767 + 1.41i)2-s + (−1.40 + 1.01i)3-s + (−0.0105 + 0.00114i)4-s + (0.435 + 1.29i)5-s + (−1.54 − 1.91i)6-s + (−0.863 + 1.27i)7-s + (0.456 + 2.78i)8-s + (0.942 − 2.84i)9-s + (−1.79 + 0.716i)10-s + (−4.10 − 0.903i)11-s + (0.0136 − 0.0122i)12-s + (1.27 + 1.08i)13-s + (−1.86 − 1.12i)14-s + (−1.92 − 1.37i)15-s + (−3.92 + 0.864i)16-s + (1.70 − 1.15i)17-s + ⋯
L(s)  = 1  + (0.0542 + 1.00i)2-s + (−0.810 + 0.585i)3-s + (−0.00526 + 0.000572i)4-s + (0.194 + 0.578i)5-s + (−0.630 − 0.779i)6-s + (−0.326 + 0.481i)7-s + (0.161 + 0.984i)8-s + (0.314 − 0.949i)9-s + (−0.568 + 0.226i)10-s + (−1.23 − 0.272i)11-s + (0.00393 − 0.00354i)12-s + (0.354 + 0.300i)13-s + (−0.499 − 0.300i)14-s + (−0.496 − 0.354i)15-s + (−0.981 + 0.216i)16-s + (0.414 − 0.281i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 - 0.459i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.888 - 0.459i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ -0.888 - 0.459i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.228564 + 0.940062i\)
\(L(\frac12)\) \(\approx\) \(0.228564 + 0.940062i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.40 - 1.01i)T \)
59 \( 1 + (-3.37 + 6.90i)T \)
good2 \( 1 + (-0.0767 - 1.41i)T + (-1.98 + 0.216i)T^{2} \)
5 \( 1 + (-0.435 - 1.29i)T + (-3.98 + 3.02i)T^{2} \)
7 \( 1 + (0.863 - 1.27i)T + (-2.59 - 6.50i)T^{2} \)
11 \( 1 + (4.10 + 0.903i)T + (9.98 + 4.61i)T^{2} \)
13 \( 1 + (-1.27 - 1.08i)T + (2.10 + 12.8i)T^{2} \)
17 \( 1 + (-1.70 + 1.15i)T + (6.29 - 15.7i)T^{2} \)
19 \( 1 + (0.106 + 0.200i)T + (-10.6 + 15.7i)T^{2} \)
23 \( 1 + (-2.98 - 2.82i)T + (1.24 + 22.9i)T^{2} \)
29 \( 1 + (1.83 + 0.0992i)T + (28.8 + 3.13i)T^{2} \)
31 \( 1 + (-5.88 - 3.11i)T + (17.3 + 25.6i)T^{2} \)
37 \( 1 + (-9.18 - 1.50i)T + (35.0 + 11.8i)T^{2} \)
41 \( 1 + (-0.553 - 0.584i)T + (-2.21 + 40.9i)T^{2} \)
43 \( 1 + (1.76 + 8.03i)T + (-39.0 + 18.0i)T^{2} \)
47 \( 1 + (-4.56 - 1.53i)T + (37.4 + 28.4i)T^{2} \)
53 \( 1 + (6.54 + 2.60i)T + (38.4 + 36.4i)T^{2} \)
61 \( 1 + (-2.05 + 0.111i)T + (60.6 - 6.59i)T^{2} \)
67 \( 1 + (15.0 - 2.47i)T + (63.4 - 21.3i)T^{2} \)
71 \( 1 + (-2.35 + 6.99i)T + (-56.5 - 42.9i)T^{2} \)
73 \( 1 + (-4.63 + 7.70i)T + (-34.1 - 64.4i)T^{2} \)
79 \( 1 + (6.67 - 3.08i)T + (51.1 - 60.2i)T^{2} \)
83 \( 1 + (-4.39 + 15.8i)T + (-71.1 - 42.7i)T^{2} \)
89 \( 1 + (-0.138 + 2.56i)T + (-88.4 - 9.62i)T^{2} \)
97 \( 1 + (3.19 + 5.31i)T + (-45.4 + 85.7i)T^{2} \)
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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

−13.23192740079584936303746177317, −11.96124084536242649039652379804, −11.03020290462328324740168562489, −10.29001217653758198242308985354, −9.036355472036307815329230195983, −7.71904818374497661612233295592, −6.59256634595696116919133263348, −5.82593406849332690416461762632, −4.89608994316215222594555603963, −2.91011497607807357927803861358, 1.01654273880555995672174488696, 2.68256396405860727997234731638, 4.47930499781563725369768393543, 5.78315758467445032348533849297, 7.00911171771339306207540883765, 8.059184613498751367937336426460, 9.744469044322839828215815020500, 10.56512805681734631068682980443, 11.24729111176973216345925389742, 12.38768160525699026960538583320

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