Properties

Label 2-177-177.104-c2-0-20
Degree $2$
Conductor $177$
Sign $0.0621 - 0.998i$
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  + (1.28 + 2.14i)2-s + (2.82 − 1.00i)3-s + (−1.04 + 1.97i)4-s + (0.792 + 7.28i)5-s + (5.79 + 4.75i)6-s + (−2.16 + 0.729i)7-s + (4.38 − 0.237i)8-s + (6.97 − 5.68i)9-s + (−14.5 + 11.0i)10-s + (−4.14 − 2.81i)11-s + (−0.974 + 6.65i)12-s + (−14.8 − 14.0i)13-s + (−4.35 − 3.69i)14-s + (9.57 + 19.7i)15-s + (11.1 + 16.5i)16-s + (5.14 − 15.2i)17-s + ⋯
L(s)  = 1  + (0.643 + 1.07i)2-s + (0.942 − 0.335i)3-s + (−0.262 + 0.494i)4-s + (0.158 + 1.45i)5-s + (0.965 + 0.792i)6-s + (−0.309 + 0.104i)7-s + (0.548 − 0.0297i)8-s + (0.775 − 0.631i)9-s + (−1.45 + 1.10i)10-s + (−0.377 − 0.255i)11-s + (−0.0811 + 0.554i)12-s + (−1.14 − 1.08i)13-s + (−0.310 − 0.263i)14-s + (0.638 + 1.31i)15-s + (0.699 + 1.03i)16-s + (0.302 − 0.897i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0621 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0621 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.0621 - 0.998i$
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: \((2,\ 177,\ (\ :1),\ 0.0621 - 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.02952 + 1.90702i\)
\(L(\frac12)\) \(\approx\) \(2.02952 + 1.90702i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.82 + 1.00i)T \)
59 \( 1 + (-58.8 - 4.55i)T \)
good2 \( 1 + (-1.28 - 2.14i)T + (-1.87 + 3.53i)T^{2} \)
5 \( 1 + (-0.792 - 7.28i)T + (-24.4 + 5.37i)T^{2} \)
7 \( 1 + (2.16 - 0.729i)T + (39.0 - 29.6i)T^{2} \)
11 \( 1 + (4.14 + 2.81i)T + (44.7 + 112. i)T^{2} \)
13 \( 1 + (14.8 + 14.0i)T + (9.14 + 168. i)T^{2} \)
17 \( 1 + (-5.14 + 15.2i)T + (-230. - 174. i)T^{2} \)
19 \( 1 + (-5.67 - 34.5i)T + (-342. + 115. i)T^{2} \)
23 \( 1 + (5.80 - 1.61i)T + (453. - 272. i)T^{2} \)
29 \( 1 + (-9.42 + 15.6i)T + (-393. - 743. i)T^{2} \)
31 \( 1 + (-4.77 + 29.1i)T + (-910. - 306. i)T^{2} \)
37 \( 1 + (-0.242 + 4.47i)T + (-1.36e3 - 148. i)T^{2} \)
41 \( 1 + (-21.6 - 6.01i)T + (1.44e3 + 866. i)T^{2} \)
43 \( 1 + (44.2 + 65.2i)T + (-684. + 1.71e3i)T^{2} \)
47 \( 1 + (-5.05 + 46.4i)T + (-2.15e3 - 474. i)T^{2} \)
53 \( 1 + (15.0 - 19.7i)T + (-751. - 2.70e3i)T^{2} \)
61 \( 1 + (21.4 - 12.8i)T + (1.74e3 - 3.28e3i)T^{2} \)
67 \( 1 + (5.18 + 95.6i)T + (-4.46e3 + 485. i)T^{2} \)
71 \( 1 + (6.89 - 63.4i)T + (-4.92e3 - 1.08e3i)T^{2} \)
73 \( 1 + (25.4 - 29.9i)T + (-862. - 5.25e3i)T^{2} \)
79 \( 1 + (30.8 - 77.3i)T + (-4.53e3 - 4.29e3i)T^{2} \)
83 \( 1 + (17.6 + 38.2i)T + (-4.45e3 + 5.25e3i)T^{2} \)
89 \( 1 + (52.8 - 87.8i)T + (-3.71e3 - 6.99e3i)T^{2} \)
97 \( 1 + (-58.3 - 68.7i)T + (-1.52e3 + 9.28e3i)T^{2} \)
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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

−13.15000044916342119647492362703, −12.02878394228776377747654128144, −10.35570572277825924549794127887, −9.864093081384711654265432445142, −7.962046635136289124247780149712, −7.48627410280150405807719358724, −6.50879264134329029330674753179, −5.47363046492247114507680089075, −3.65009515569979290453650921493, −2.51969052746885389746495001338, 1.64980250915925369018763185751, 2.95344521607195461717187293039, 4.47798399317719105986779423807, 4.88833537965397560917049444351, 7.18193702148196056877301128092, 8.429189902082052313585928224013, 9.401610667465855173725973303090, 10.12423447771165210849471914483, 11.44019228306188194875961034066, 12.57015414015738729813422417627

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