Properties

Label 2-177-177.104-c2-0-12
Degree $2$
Conductor $177$
Sign $-0.929 + 0.369i$
Analytic cond. $4.82290$
Root an. cond. $2.19611$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.90 + 3.16i)2-s + (−2.63 + 1.42i)3-s + (−4.50 + 8.50i)4-s + (0.853 + 7.84i)5-s + (−9.53 − 5.62i)6-s + (12.1 − 4.09i)7-s + (−20.7 + 1.12i)8-s + (4.92 − 7.53i)9-s + (−23.1 + 17.6i)10-s + (−2.40 − 1.62i)11-s + (−0.247 − 28.8i)12-s + (−1.30 − 1.23i)13-s + (36.0 + 30.6i)14-s + (−13.4 − 19.4i)15-s + (−21.4 − 31.6i)16-s + (5.27 − 15.6i)17-s + ⋯
L(s)  = 1  + (0.951 + 1.58i)2-s + (−0.879 + 0.475i)3-s + (−1.12 + 2.12i)4-s + (0.170 + 1.56i)5-s + (−1.58 − 0.938i)6-s + (1.73 − 0.584i)7-s + (−2.59 + 0.140i)8-s + (0.546 − 0.837i)9-s + (−2.31 + 1.76i)10-s + (−0.218 − 0.148i)11-s + (−0.0206 − 2.40i)12-s + (−0.100 − 0.0949i)13-s + (2.57 + 2.18i)14-s + (−0.896 − 1.29i)15-s + (−1.33 − 1.97i)16-s + (0.310 − 0.920i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.929 + 0.369i$
Analytic conductor: \(4.82290\)
Root analytic conductor: \(2.19611\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1),\ -0.929 + 0.369i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.371502 - 1.93759i\)
\(L(\frac12)\) \(\approx\) \(0.371502 - 1.93759i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.63 - 1.42i)T \)
59 \( 1 + (-12.2 - 57.7i)T \)
good2 \( 1 + (-1.90 - 3.16i)T + (-1.87 + 3.53i)T^{2} \)
5 \( 1 + (-0.853 - 7.84i)T + (-24.4 + 5.37i)T^{2} \)
7 \( 1 + (-12.1 + 4.09i)T + (39.0 - 29.6i)T^{2} \)
11 \( 1 + (2.40 + 1.62i)T + (44.7 + 112. i)T^{2} \)
13 \( 1 + (1.30 + 1.23i)T + (9.14 + 168. i)T^{2} \)
17 \( 1 + (-5.27 + 15.6i)T + (-230. - 174. i)T^{2} \)
19 \( 1 + (0.0242 + 0.148i)T + (-342. + 115. i)T^{2} \)
23 \( 1 + (-30.3 + 8.42i)T + (453. - 272. i)T^{2} \)
29 \( 1 + (1.00 - 1.66i)T + (-393. - 743. i)T^{2} \)
31 \( 1 + (6.01 - 36.6i)T + (-910. - 306. i)T^{2} \)
37 \( 1 + (1.35 - 24.9i)T + (-1.36e3 - 148. i)T^{2} \)
41 \( 1 + (64.5 + 17.9i)T + (1.44e3 + 866. i)T^{2} \)
43 \( 1 + (19.8 + 29.2i)T + (-684. + 1.71e3i)T^{2} \)
47 \( 1 + (-4.02 + 37.0i)T + (-2.15e3 - 474. i)T^{2} \)
53 \( 1 + (-18.5 + 24.4i)T + (-751. - 2.70e3i)T^{2} \)
61 \( 1 + (-39.1 + 23.5i)T + (1.74e3 - 3.28e3i)T^{2} \)
67 \( 1 + (2.79 + 51.5i)T + (-4.46e3 + 485. i)T^{2} \)
71 \( 1 + (-2.48 + 22.8i)T + (-4.92e3 - 1.08e3i)T^{2} \)
73 \( 1 + (20.7 - 24.3i)T + (-862. - 5.25e3i)T^{2} \)
79 \( 1 + (23.3 - 58.6i)T + (-4.53e3 - 4.29e3i)T^{2} \)
83 \( 1 + (-9.41 - 20.3i)T + (-4.45e3 + 5.25e3i)T^{2} \)
89 \( 1 + (22.7 - 37.8i)T + (-3.71e3 - 6.99e3i)T^{2} \)
97 \( 1 + (-0.456 - 0.537i)T + (-1.52e3 + 9.28e3i)T^{2} \)
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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

−13.53148919226304952353081396622, −12.00132926324323937273502078474, −11.14967585311126743450331717315, −10.29497184736777534055910094116, −8.493625766069237575534388107939, −7.16967175412834329417328503724, −6.88925633543185483344380753521, −5.42393702098264772935350351939, −4.79281361028479700514321234555, −3.41324721099811065175115559742, 1.15149636546473640097814206336, 1.94802160223955241161650084957, 4.40560055749878163601531839778, 5.08178340970432186914459928715, 5.69415135837553781439226537224, 8.015802688179970617316979672698, 9.116288064104726259940746115737, 10.40926131737383266510158431964, 11.45510200433307397761660416624, 11.81501079060920962026951979916

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