Properties

Label 2-177-177.2-c3-0-8
Degree $2$
Conductor $177$
Sign $0.736 + 0.675i$
Analytic cond. $10.4433$
Root an. cond. $3.23161$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.282 + 5.20i)2-s + (−0.412 + 5.17i)3-s + (−19.0 − 2.07i)4-s + (−2.66 + 7.89i)5-s + (−26.8 − 3.61i)6-s + (2.27 + 3.35i)7-s + (9.45 − 57.6i)8-s + (−26.6 − 4.27i)9-s + (−40.3 − 16.0i)10-s + (19.5 − 4.29i)11-s + (18.6 − 98.0i)12-s + (−36.1 + 30.6i)13-s + (−18.1 + 10.9i)14-s + (−39.7 − 17.0i)15-s + (147. + 32.4i)16-s + (48.1 + 32.6i)17-s + ⋯
L(s)  = 1  + (−0.0998 + 1.84i)2-s + (−0.0794 + 0.996i)3-s + (−2.38 − 0.259i)4-s + (−0.237 + 0.706i)5-s + (−1.82 − 0.245i)6-s + (0.122 + 0.181i)7-s + (0.417 − 2.54i)8-s + (−0.987 − 0.158i)9-s + (−1.27 − 0.508i)10-s + (0.535 − 0.117i)11-s + (0.448 − 2.35i)12-s + (−0.770 + 0.654i)13-s + (−0.346 + 0.208i)14-s + (−0.685 − 0.293i)15-s + (2.30 + 0.507i)16-s + (0.686 + 0.465i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 + 0.675i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.736 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.736 + 0.675i$
Analytic conductor: \(10.4433\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ 0.736 + 0.675i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.657165 - 0.255719i\)
\(L(\frac12)\) \(\approx\) \(0.657165 - 0.255719i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.412 - 5.17i)T \)
59 \( 1 + (116. + 437. i)T \)
good2 \( 1 + (0.282 - 5.20i)T + (-7.95 - 0.864i)T^{2} \)
5 \( 1 + (2.66 - 7.89i)T + (-99.5 - 75.6i)T^{2} \)
7 \( 1 + (-2.27 - 3.35i)T + (-126. + 318. i)T^{2} \)
11 \( 1 + (-19.5 + 4.29i)T + (1.20e3 - 558. i)T^{2} \)
13 \( 1 + (36.1 - 30.6i)T + (355. - 2.16e3i)T^{2} \)
17 \( 1 + (-48.1 - 32.6i)T + (1.81e3 + 4.56e3i)T^{2} \)
19 \( 1 + (-39.6 + 74.8i)T + (-3.84e3 - 5.67e3i)T^{2} \)
23 \( 1 + (107. - 101. i)T + (658. - 1.21e4i)T^{2} \)
29 \( 1 + (-71.3 + 3.86i)T + (2.42e4 - 2.63e3i)T^{2} \)
31 \( 1 + (242. - 128. i)T + (1.67e4 - 2.46e4i)T^{2} \)
37 \( 1 + (40.4 - 6.62i)T + (4.80e4 - 1.61e4i)T^{2} \)
41 \( 1 + (-224. + 237. i)T + (-3.73e3 - 6.88e4i)T^{2} \)
43 \( 1 + (76.7 - 348. i)T + (-7.21e4 - 3.33e4i)T^{2} \)
47 \( 1 + (-136. + 46.0i)T + (8.26e4 - 6.28e4i)T^{2} \)
53 \( 1 + (251. - 100. i)T + (1.08e5 - 1.02e5i)T^{2} \)
61 \( 1 + (778. + 42.1i)T + (2.25e5 + 2.45e4i)T^{2} \)
67 \( 1 + (-336. - 55.1i)T + (2.85e5 + 9.60e4i)T^{2} \)
71 \( 1 + (134. + 399. i)T + (-2.84e5 + 2.16e5i)T^{2} \)
73 \( 1 + (260. + 432. i)T + (-1.82e5 + 3.43e5i)T^{2} \)
79 \( 1 + (389. + 180. i)T + (3.19e5 + 3.75e5i)T^{2} \)
83 \( 1 + (-57.4 - 207. i)T + (-4.89e5 + 2.94e5i)T^{2} \)
89 \( 1 + (-24.4 - 451. i)T + (-7.00e5 + 7.62e4i)T^{2} \)
97 \( 1 + (652. - 1.08e3i)T + (-4.27e5 - 8.06e5i)T^{2} \)
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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

−13.78991774182425744316791458428, −12.12818560410598682042324174650, −10.89511944935331100970695107991, −9.629102660690033443625136338255, −9.034516218253526264896954213072, −7.80732712555337361393179032061, −6.86242064697783753822796339434, −5.76774379087857181561420559061, −4.78004799233405470392681747664, −3.56220124905143294346455979907, 0.35821960963468708726535507937, 1.48014046069720438630726771546, 2.84184432286393637019576904345, 4.29134420346433489637972914937, 5.59805580345081940031835802184, 7.57119494155261810974513776144, 8.492096478033991111173817543542, 9.548263785259335854492461983510, 10.55209711317935077444836067874, 11.66921470283010219808628594842

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