Properties

Label 2-177-177.23-c1-0-4
Degree $2$
Conductor $177$
Sign $-0.0803 - 0.996i$
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  + (−0.0604 + 0.0572i)2-s + (1.46 + 0.930i)3-s + (−0.107 + 1.99i)4-s + (−3.23 − 0.529i)5-s + (−0.141 + 0.0273i)6-s + (−1.62 + 3.06i)7-s + (−0.215 − 0.253i)8-s + (1.26 + 2.71i)9-s + (0.225 − 0.152i)10-s + (3.14 − 0.342i)11-s + (−2.01 + 2.80i)12-s + (1.78 − 3.86i)13-s + (−0.0772 − 0.278i)14-s + (−4.22 − 3.78i)15-s + (−3.93 − 0.427i)16-s + (1.00 − 0.534i)17-s + ⋯
L(s)  = 1  + (−0.0427 + 0.0404i)2-s + (0.843 + 0.537i)3-s + (−0.0539 + 0.995i)4-s + (−1.44 − 0.237i)5-s + (−0.0577 + 0.0111i)6-s + (−0.614 + 1.15i)7-s + (−0.0760 − 0.0895i)8-s + (0.422 + 0.906i)9-s + (0.0713 − 0.0483i)10-s + (0.949 − 0.103i)11-s + (−0.580 + 0.810i)12-s + (0.495 − 1.07i)13-s + (−0.0206 − 0.0743i)14-s + (−1.09 − 0.976i)15-s + (−0.983 − 0.106i)16-s + (0.244 − 0.129i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0803 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0803 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.0803 - 0.996i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ -0.0803 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.775215 + 0.840231i\)
\(L(\frac12)\) \(\approx\) \(0.775215 + 0.840231i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.46 - 0.930i)T \)
59 \( 1 + (-4.21 + 6.42i)T \)
good2 \( 1 + (0.0604 - 0.0572i)T + (0.108 - 1.99i)T^{2} \)
5 \( 1 + (3.23 + 0.529i)T + (4.73 + 1.59i)T^{2} \)
7 \( 1 + (1.62 - 3.06i)T + (-3.92 - 5.79i)T^{2} \)
11 \( 1 + (-3.14 + 0.342i)T + (10.7 - 2.36i)T^{2} \)
13 \( 1 + (-1.78 + 3.86i)T + (-8.41 - 9.90i)T^{2} \)
17 \( 1 + (-1.00 + 0.534i)T + (9.54 - 14.0i)T^{2} \)
19 \( 1 + (-5.72 + 3.44i)T + (8.89 - 16.7i)T^{2} \)
23 \( 1 + (-3.32 - 8.34i)T + (-16.6 + 15.8i)T^{2} \)
29 \( 1 + (-3.45 + 3.64i)T + (-1.57 - 28.9i)T^{2} \)
31 \( 1 + (1.85 - 3.09i)T + (-14.5 - 27.3i)T^{2} \)
37 \( 1 + (3.97 + 3.37i)T + (5.98 + 36.5i)T^{2} \)
41 \( 1 + (7.19 + 2.86i)T + (29.7 + 28.1i)T^{2} \)
43 \( 1 + (0.350 - 3.22i)T + (-41.9 - 9.24i)T^{2} \)
47 \( 1 + (0.457 + 2.79i)T + (-44.5 + 15.0i)T^{2} \)
53 \( 1 + (3.98 + 2.69i)T + (19.6 + 49.2i)T^{2} \)
61 \( 1 + (1.06 + 1.12i)T + (-3.30 + 60.9i)T^{2} \)
67 \( 1 + (3.77 - 3.20i)T + (10.8 - 66.1i)T^{2} \)
71 \( 1 + (-6.29 + 1.03i)T + (67.2 - 22.6i)T^{2} \)
73 \( 1 + (-15.0 + 4.16i)T + (62.5 - 37.6i)T^{2} \)
79 \( 1 + (4.05 + 0.892i)T + (71.6 + 33.1i)T^{2} \)
83 \( 1 + (5.19 + 3.95i)T + (22.2 + 79.9i)T^{2} \)
89 \( 1 + (3.52 + 3.33i)T + (4.81 + 88.8i)T^{2} \)
97 \( 1 + (-4.94 - 1.37i)T + (83.1 + 50.0i)T^{2} \)
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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

−12.84719717202830249531633771790, −11.95603129311504755141098746525, −11.30748454871335477726748920107, −9.532415109978762032626828874303, −8.807212632239095601841487759853, −8.037549115397743718567615462164, −7.13300789227178823784894136575, −5.13813095309240376647989074176, −3.57887558893511847821148727141, −3.18342576320835198336145557103, 1.12462133436482515561373217613, 3.45265400066772091264884523006, 4.32863056428157758639587659297, 6.61937060074990568499784561815, 7.04419075890136514461399493202, 8.335577433603649421206326428822, 9.379942599424661663811826614216, 10.39989287249017994315962791026, 11.50486508758150238602293828835, 12.36805403548857681866293481011

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