Properties

Label 2-177-59.17-c1-0-5
Degree $2$
Conductor $177$
Sign $0.842 + 0.538i$
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  + (1.03 − 1.52i)2-s + (0.0541 + 0.998i)3-s + (−0.512 − 1.28i)4-s + (1.77 + 0.819i)5-s + (1.57 + 0.948i)6-s + (0.611 + 2.20i)7-s + (1.10 + 0.243i)8-s + (−0.994 + 0.108i)9-s + (3.07 − 1.85i)10-s + (−4.59 − 4.35i)11-s + (1.25 − 0.581i)12-s + (−4.53 − 0.493i)13-s + (3.98 + 1.34i)14-s + (−0.721 + 1.81i)15-s + (3.52 − 3.33i)16-s + (1.27 − 4.59i)17-s + ⋯
L(s)  = 1  + (0.730 − 1.07i)2-s + (0.0312 + 0.576i)3-s + (−0.256 − 0.643i)4-s + (0.791 + 0.366i)5-s + (0.643 + 0.387i)6-s + (0.230 + 0.831i)7-s + (0.390 + 0.0859i)8-s + (−0.331 + 0.0360i)9-s + (0.972 − 0.585i)10-s + (−1.38 − 1.31i)11-s + (0.362 − 0.167i)12-s + (−1.25 − 0.136i)13-s + (1.06 + 0.358i)14-s + (−0.186 + 0.467i)15-s + (0.880 − 0.834i)16-s + (0.309 − 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 + 0.538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.842 + 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.842 + 0.538i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 0.842 + 0.538i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71532 - 0.501197i\)
\(L(\frac12)\) \(\approx\) \(1.71532 - 0.501197i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0541 - 0.998i)T \)
59 \( 1 + (6.46 - 4.15i)T \)
good2 \( 1 + (-1.03 + 1.52i)T + (-0.740 - 1.85i)T^{2} \)
5 \( 1 + (-1.77 - 0.819i)T + (3.23 + 3.81i)T^{2} \)
7 \( 1 + (-0.611 - 2.20i)T + (-5.99 + 3.60i)T^{2} \)
11 \( 1 + (4.59 + 4.35i)T + (0.595 + 10.9i)T^{2} \)
13 \( 1 + (4.53 + 0.493i)T + (12.6 + 2.79i)T^{2} \)
17 \( 1 + (-1.27 + 4.59i)T + (-14.5 - 8.76i)T^{2} \)
19 \( 1 + (2.06 + 1.56i)T + (5.08 + 18.3i)T^{2} \)
23 \( 1 + (-4.04 - 7.62i)T + (-12.9 + 19.0i)T^{2} \)
29 \( 1 + (-3.13 - 4.62i)T + (-10.7 + 26.9i)T^{2} \)
31 \( 1 + (1.72 - 1.31i)T + (8.29 - 29.8i)T^{2} \)
37 \( 1 + (6.46 - 1.42i)T + (33.5 - 15.5i)T^{2} \)
41 \( 1 + (-5.00 + 9.43i)T + (-23.0 - 33.9i)T^{2} \)
43 \( 1 + (3.92 - 3.71i)T + (2.32 - 42.9i)T^{2} \)
47 \( 1 + (3.31 - 1.53i)T + (30.4 - 35.8i)T^{2} \)
53 \( 1 + (-7.88 - 4.74i)T + (24.8 + 46.8i)T^{2} \)
61 \( 1 + (1.13 - 1.67i)T + (-22.5 - 56.6i)T^{2} \)
67 \( 1 + (-5.09 - 1.12i)T + (60.8 + 28.1i)T^{2} \)
71 \( 1 + (-0.960 + 0.444i)T + (45.9 - 54.1i)T^{2} \)
73 \( 1 + (-2.51 - 0.846i)T + (58.1 + 44.1i)T^{2} \)
79 \( 1 + (0.341 - 6.29i)T + (-78.5 - 8.54i)T^{2} \)
83 \( 1 + (-0.833 + 5.08i)T + (-78.6 - 26.5i)T^{2} \)
89 \( 1 + (2.24 + 3.31i)T + (-32.9 + 82.6i)T^{2} \)
97 \( 1 + (-3.57 + 1.20i)T + (77.2 - 58.7i)T^{2} \)
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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

−12.52962795195294387487364209971, −11.58151690773902584939051046073, −10.74418038603660038488593524968, −9.998517141347438602676118470600, −8.881742417803374305528976308073, −7.46113293287153518095527171987, −5.48351624945660191931178675424, −5.09214189276165071929511827142, −3.14021639674122256252444752783, −2.45969055985213003507940096177, 2.06597457007372002780395657170, 4.48793965336468952829524536120, 5.27565663190385905921192004825, 6.51011625140340784311073393735, 7.39784589941012631876375233606, 8.171267797569844115939744002716, 9.945961767966179174021150864839, 10.56067271851109806800066797321, 12.52442582579123839777627344669, 12.89405231189375232528486895773

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