Properties

Label 2-177-59.51-c1-0-7
Degree $2$
Conductor $177$
Sign $-0.643 + 0.765i$
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  + (0.276 − 1.68i)2-s + (−0.267 + 0.963i)3-s + (−0.866 − 0.292i)4-s + (−2.40 − 3.55i)5-s + (1.54 + 0.716i)6-s + (0.331 − 0.0728i)7-s + (0.867 − 1.63i)8-s + (−0.856 − 0.515i)9-s + (−6.65 + 3.07i)10-s + (0.00152 − 0.00116i)11-s + (0.513 − 0.757i)12-s + (1.61 − 0.969i)13-s + (−0.0313 − 0.578i)14-s + (4.06 − 1.37i)15-s + (−3.97 − 3.02i)16-s + (−1.13 − 0.250i)17-s + ⋯
L(s)  = 1  + (0.195 − 1.19i)2-s + (−0.154 + 0.556i)3-s + (−0.433 − 0.146i)4-s + (−1.07 − 1.58i)5-s + (0.632 + 0.292i)6-s + (0.125 − 0.0275i)7-s + (0.306 − 0.578i)8-s + (−0.285 − 0.171i)9-s + (−2.10 + 0.973i)10-s + (0.000461 − 0.000350i)11-s + (0.148 − 0.218i)12-s + (0.447 − 0.269i)13-s + (−0.00837 − 0.154i)14-s + (1.05 − 0.353i)15-s + (−0.993 − 0.755i)16-s + (−0.276 − 0.0607i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.643 + 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.643 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.643 + 0.765i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ -0.643 + 0.765i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.458612 - 0.984870i\)
\(L(\frac12)\) \(\approx\) \(0.458612 - 0.984870i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.267 - 0.963i)T \)
59 \( 1 + (7.10 + 2.91i)T \)
good2 \( 1 + (-0.276 + 1.68i)T + (-1.89 - 0.638i)T^{2} \)
5 \( 1 + (2.40 + 3.55i)T + (-1.85 + 4.64i)T^{2} \)
7 \( 1 + (-0.331 + 0.0728i)T + (6.35 - 2.93i)T^{2} \)
11 \( 1 + (-0.00152 + 0.00116i)T + (2.94 - 10.5i)T^{2} \)
13 \( 1 + (-1.61 + 0.969i)T + (6.08 - 11.4i)T^{2} \)
17 \( 1 + (1.13 + 0.250i)T + (15.4 + 7.13i)T^{2} \)
19 \( 1 + (-3.48 + 0.379i)T + (18.5 - 4.08i)T^{2} \)
23 \( 1 + (-2.65 - 3.12i)T + (-3.72 + 22.6i)T^{2} \)
29 \( 1 + (-1.38 - 8.45i)T + (-27.4 + 9.25i)T^{2} \)
31 \( 1 + (-9.43 - 1.02i)T + (30.2 + 6.66i)T^{2} \)
37 \( 1 + (3.16 + 5.96i)T + (-20.7 + 30.6i)T^{2} \)
41 \( 1 + (-6.48 + 7.63i)T + (-6.63 - 40.4i)T^{2} \)
43 \( 1 + (1.69 + 1.28i)T + (11.5 + 41.4i)T^{2} \)
47 \( 1 + (-0.206 + 0.304i)T + (-17.3 - 43.6i)T^{2} \)
53 \( 1 + (-9.03 - 4.17i)T + (34.3 + 40.3i)T^{2} \)
61 \( 1 + (-0.0632 + 0.385i)T + (-57.8 - 19.4i)T^{2} \)
67 \( 1 + (-2.23 + 4.21i)T + (-37.5 - 55.4i)T^{2} \)
71 \( 1 + (7.04 - 10.3i)T + (-26.2 - 65.9i)T^{2} \)
73 \( 1 + (0.205 + 3.78i)T + (-72.5 + 7.89i)T^{2} \)
79 \( 1 + (-1.63 - 5.89i)T + (-67.6 + 40.7i)T^{2} \)
83 \( 1 + (6.90 + 6.53i)T + (4.49 + 82.8i)T^{2} \)
89 \( 1 + (-1.15 - 7.04i)T + (-84.3 + 28.4i)T^{2} \)
97 \( 1 + (0.336 - 6.20i)T + (-96.4 - 10.4i)T^{2} \)
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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

−12.19499962062757230298287171823, −11.48126889344879672656987279038, −10.66117337190392099471248440312, −9.395292067671562173081408268379, −8.614486550964678275453868950947, −7.34898103907146588783505761499, −5.26988649726246686497808789258, −4.35375072628977349083197448090, −3.34982980677647904590587550017, −1.06797449307426655633071350997, 2.79003483752054182539712189688, 4.45719388125705368332401524343, 6.19008666378951487974180476102, 6.72903108480410037065363320010, 7.67690064964123995700373545368, 8.312721645796024951609238809947, 10.24849341648806196263045550687, 11.36385712928596377368622968792, 11.75921859815113306553335524375, 13.46651869964905858832862539206

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