Properties

Label 2-177-177.23-c1-0-2
Degree $2$
Conductor $177$
Sign $-0.771 - 0.635i$
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  + (−0.703 + 0.666i)2-s + (−1.32 − 1.11i)3-s + (−0.0574 + 1.05i)4-s + (0.383 + 0.0629i)5-s + (1.67 − 0.0973i)6-s + (−0.729 + 1.37i)7-s + (−1.92 − 2.26i)8-s + (0.508 + 2.95i)9-s + (−0.311 + 0.211i)10-s + (−2.37 + 0.258i)11-s + (1.25 − 1.33i)12-s + (−2.83 + 6.13i)13-s + (−0.403 − 1.45i)14-s + (−0.438 − 0.511i)15-s + (0.747 + 0.0812i)16-s + (−5.86 + 3.10i)17-s + ⋯
L(s)  = 1  + (−0.497 + 0.471i)2-s + (−0.764 − 0.644i)3-s + (−0.0287 + 0.529i)4-s + (0.171 + 0.0281i)5-s + (0.683 − 0.0397i)6-s + (−0.275 + 0.520i)7-s + (−0.678 − 0.799i)8-s + (0.169 + 0.985i)9-s + (−0.0986 + 0.0668i)10-s + (−0.716 + 0.0779i)11-s + (0.363 − 0.386i)12-s + (−0.787 + 1.70i)13-s + (−0.107 − 0.388i)14-s + (−0.113 − 0.132i)15-s + (0.186 + 0.0203i)16-s + (−1.42 + 0.753i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 - 0.635i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.771 - 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.771 - 0.635i$
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: \((2,\ 177,\ (\ :1/2),\ -0.771 - 0.635i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.143837 + 0.400776i\)
\(L(\frac12)\) \(\approx\) \(0.143837 + 0.400776i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.32 + 1.11i)T \)
59 \( 1 + (-2.66 - 7.20i)T \)
good2 \( 1 + (0.703 - 0.666i)T + (0.108 - 1.99i)T^{2} \)
5 \( 1 + (-0.383 - 0.0629i)T + (4.73 + 1.59i)T^{2} \)
7 \( 1 + (0.729 - 1.37i)T + (-3.92 - 5.79i)T^{2} \)
11 \( 1 + (2.37 - 0.258i)T + (10.7 - 2.36i)T^{2} \)
13 \( 1 + (2.83 - 6.13i)T + (-8.41 - 9.90i)T^{2} \)
17 \( 1 + (5.86 - 3.10i)T + (9.54 - 14.0i)T^{2} \)
19 \( 1 + (-4.92 + 2.96i)T + (8.89 - 16.7i)T^{2} \)
23 \( 1 + (-2.59 - 6.50i)T + (-16.6 + 15.8i)T^{2} \)
29 \( 1 + (-1.89 + 1.99i)T + (-1.57 - 28.9i)T^{2} \)
31 \( 1 + (-4.19 + 6.96i)T + (-14.5 - 27.3i)T^{2} \)
37 \( 1 + (-3.34 - 2.84i)T + (5.98 + 36.5i)T^{2} \)
41 \( 1 + (4.48 + 1.78i)T + (29.7 + 28.1i)T^{2} \)
43 \( 1 + (0.462 - 4.25i)T + (-41.9 - 9.24i)T^{2} \)
47 \( 1 + (0.416 + 2.54i)T + (-44.5 + 15.0i)T^{2} \)
53 \( 1 + (-5.01 - 3.39i)T + (19.6 + 49.2i)T^{2} \)
61 \( 1 + (2.11 + 2.23i)T + (-3.30 + 60.9i)T^{2} \)
67 \( 1 + (-2.81 + 2.39i)T + (10.8 - 66.1i)T^{2} \)
71 \( 1 + (2.93 - 0.481i)T + (67.2 - 22.6i)T^{2} \)
73 \( 1 + (-1.02 + 0.285i)T + (62.5 - 37.6i)T^{2} \)
79 \( 1 + (9.43 + 2.07i)T + (71.6 + 33.1i)T^{2} \)
83 \( 1 + (9.05 + 6.88i)T + (22.2 + 79.9i)T^{2} \)
89 \( 1 + (-2.33 - 2.21i)T + (4.81 + 88.8i)T^{2} \)
97 \( 1 + (-4.04 - 1.12i)T + (83.1 + 50.0i)T^{2} \)
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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

−13.14224250935833062739429654026, −11.87265110631348363392628541322, −11.51527653923238819886659175891, −9.861339243628991205287689181050, −8.985481371157401402702574025686, −7.73305008845218336847099903670, −6.93251300790724451455322391498, −6.00196125227045795739786893984, −4.51365399579518496945342196279, −2.38414532498663392022239252609, 0.47944825200014068135383546600, 2.91539685234450265942534351998, 4.86062538170935422840695828055, 5.62630710415960470551774119001, 6.97778616708180173964510928489, 8.527273663609985931436741698075, 9.808720522805211881829542710604, 10.25633574808650057503318579970, 10.98695375052177838170127400314, 12.04236745892773080999091117151

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