Properties

Label 2-177-59.20-c1-0-2
Degree $2$
Conductor $177$
Sign $0.796 + 0.604i$
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  + (−2.13 − 0.988i)2-s + (0.947 − 0.319i)3-s + (2.29 + 2.70i)4-s + (2.84 − 1.71i)5-s + (−2.34 − 0.254i)6-s + (0.205 + 3.79i)7-s + (−0.972 − 3.50i)8-s + (0.796 − 0.605i)9-s + (−7.78 + 0.846i)10-s + (0.766 + 4.67i)11-s + (3.03 + 1.82i)12-s + (1.32 + 1.00i)13-s + (3.31 − 8.32i)14-s + (2.15 − 2.53i)15-s + (−0.238 + 1.45i)16-s + (0.334 − 6.17i)17-s + ⋯
L(s)  = 1  + (−1.51 − 0.699i)2-s + (0.547 − 0.184i)3-s + (1.14 + 1.35i)4-s + (1.27 − 0.766i)5-s + (−0.955 − 0.103i)6-s + (0.0778 + 1.43i)7-s + (−0.344 − 1.23i)8-s + (0.265 − 0.201i)9-s + (−2.46 + 0.267i)10-s + (0.231 + 1.41i)11-s + (0.876 + 0.527i)12-s + (0.368 + 0.279i)13-s + (0.886 − 2.22i)14-s + (0.555 − 0.654i)15-s + (−0.0596 + 0.363i)16-s + (0.0812 − 1.49i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.796 + 0.604i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 0.796 + 0.604i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.796934 - 0.268098i\)
\(L(\frac12)\) \(\approx\) \(0.796934 - 0.268098i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.947 + 0.319i)T \)
59 \( 1 + (-5.49 - 5.36i)T \)
good2 \( 1 + (2.13 + 0.988i)T + (1.29 + 1.52i)T^{2} \)
5 \( 1 + (-2.84 + 1.71i)T + (2.34 - 4.41i)T^{2} \)
7 \( 1 + (-0.205 - 3.79i)T + (-6.95 + 0.756i)T^{2} \)
11 \( 1 + (-0.766 - 4.67i)T + (-10.4 + 3.51i)T^{2} \)
13 \( 1 + (-1.32 - 1.00i)T + (3.47 + 12.5i)T^{2} \)
17 \( 1 + (-0.334 + 6.17i)T + (-16.9 - 1.83i)T^{2} \)
19 \( 1 + (0.760 + 0.720i)T + (1.02 + 18.9i)T^{2} \)
23 \( 1 + (6.59 + 1.45i)T + (20.8 + 9.65i)T^{2} \)
29 \( 1 + (-1.47 + 0.682i)T + (18.7 - 22.1i)T^{2} \)
31 \( 1 + (-3.36 + 3.18i)T + (1.67 - 30.9i)T^{2} \)
37 \( 1 + (-0.453 + 1.63i)T + (-31.7 - 19.0i)T^{2} \)
41 \( 1 + (5.60 - 1.23i)T + (37.2 - 17.2i)T^{2} \)
43 \( 1 + (0.677 - 4.13i)T + (-40.7 - 13.7i)T^{2} \)
47 \( 1 + (11.1 + 6.72i)T + (22.0 + 41.5i)T^{2} \)
53 \( 1 + (3.87 + 0.421i)T + (51.7 + 11.3i)T^{2} \)
61 \( 1 + (0.386 + 0.178i)T + (39.4 + 46.4i)T^{2} \)
67 \( 1 + (4.22 + 15.2i)T + (-57.4 + 34.5i)T^{2} \)
71 \( 1 + (-9.90 - 5.95i)T + (33.2 + 62.7i)T^{2} \)
73 \( 1 + (1.94 - 4.88i)T + (-52.9 - 50.2i)T^{2} \)
79 \( 1 + (5.10 + 1.71i)T + (62.8 + 47.8i)T^{2} \)
83 \( 1 + (-3.34 - 4.93i)T + (-30.7 + 77.1i)T^{2} \)
89 \( 1 + (-9.08 + 4.20i)T + (57.6 - 67.8i)T^{2} \)
97 \( 1 + (-1.24 - 3.12i)T + (-70.4 + 66.7i)T^{2} \)
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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

−12.24096782074528804467828430681, −11.70609335404934898242595016627, −9.966035245751161706462986298953, −9.559249966254478020739805198658, −8.897146419764009599749098404241, −8.002755719446202843741619355422, −6.53726278945850240480390739375, −5.02077351944593471664830146006, −2.47920233666009031722804972152, −1.78310516945601563590333743751, 1.53339253753629008087744471750, 3.58621913525830404786086646478, 6.01254489232387686433811295928, 6.63128271795271252679388319343, 7.927884819442423485409971013490, 8.603153277759226959229490461069, 9.942835134253638691582108143697, 10.31274684686365679042413683071, 11.02901045231584600097474859805, 13.26342650064221924166096991238

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