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Properties

Label 2-177-59.51-c1-0-3
Degree $2$
Conductor $177$
Sign $0.501 - 0.865i$
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  + (−0.0772 + 0.470i)2-s + (−0.267 + 0.963i)3-s + (1.67 + 0.565i)4-s + (−0.369 − 0.544i)5-s + (−0.433 − 0.200i)6-s + (3.18 − 0.700i)7-s + (−0.843 + 1.59i)8-s + (−0.856 − 0.515i)9-s + (0.284 − 0.131i)10-s + (−1.46 + 1.11i)11-s + (−0.994 + 1.46i)12-s + (−2.36 + 1.42i)13-s + (0.0841 + 1.55i)14-s + (0.623 − 0.210i)15-s + (2.13 + 1.62i)16-s + (4.71 + 1.03i)17-s + ⋯
L(s)  = 1  + (−0.0545 + 0.332i)2-s + (−0.154 + 0.556i)3-s + (0.839 + 0.282i)4-s + (−0.165 − 0.243i)5-s + (−0.176 − 0.0818i)6-s + (1.20 − 0.264i)7-s + (−0.298 + 0.562i)8-s + (−0.285 − 0.171i)9-s + (0.0900 − 0.0416i)10-s + (−0.440 + 0.335i)11-s + (−0.287 + 0.423i)12-s + (−0.655 + 0.394i)13-s + (0.0224 + 0.414i)14-s + (0.160 − 0.0542i)15-s + (0.534 + 0.406i)16-s + (1.14 + 0.251i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.501 - 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.501 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.501 - 0.865i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 0.501 - 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13346 + 0.652974i\)
\(L(\frac12)\) \(\approx\) \(1.13346 + 0.652974i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.267 - 0.963i)T \)
59 \( 1 + (-7.50 - 1.64i)T \)
good2 \( 1 + (0.0772 - 0.470i)T + (-1.89 - 0.638i)T^{2} \)
5 \( 1 + (0.369 + 0.544i)T + (-1.85 + 4.64i)T^{2} \)
7 \( 1 + (-3.18 + 0.700i)T + (6.35 - 2.93i)T^{2} \)
11 \( 1 + (1.46 - 1.11i)T + (2.94 - 10.5i)T^{2} \)
13 \( 1 + (2.36 - 1.42i)T + (6.08 - 11.4i)T^{2} \)
17 \( 1 + (-4.71 - 1.03i)T + (15.4 + 7.13i)T^{2} \)
19 \( 1 + (3.87 - 0.420i)T + (18.5 - 4.08i)T^{2} \)
23 \( 1 + (5.38 + 6.34i)T + (-3.72 + 22.6i)T^{2} \)
29 \( 1 + (-0.0667 - 0.407i)T + (-27.4 + 9.25i)T^{2} \)
31 \( 1 + (-4.84 - 0.526i)T + (30.2 + 6.66i)T^{2} \)
37 \( 1 + (2.50 + 4.71i)T + (-20.7 + 30.6i)T^{2} \)
41 \( 1 + (-4.27 + 5.02i)T + (-6.63 - 40.4i)T^{2} \)
43 \( 1 + (6.45 + 4.90i)T + (11.5 + 41.4i)T^{2} \)
47 \( 1 + (0.180 - 0.265i)T + (-17.3 - 43.6i)T^{2} \)
53 \( 1 + (8.04 + 3.72i)T + (34.3 + 40.3i)T^{2} \)
61 \( 1 + (-0.932 + 5.68i)T + (-57.8 - 19.4i)T^{2} \)
67 \( 1 + (3.98 - 7.52i)T + (-37.5 - 55.4i)T^{2} \)
71 \( 1 + (3.38 - 4.98i)T + (-26.2 - 65.9i)T^{2} \)
73 \( 1 + (0.228 + 4.21i)T + (-72.5 + 7.89i)T^{2} \)
79 \( 1 + (-4.10 - 14.7i)T + (-67.6 + 40.7i)T^{2} \)
83 \( 1 + (-11.6 - 11.0i)T + (4.49 + 82.8i)T^{2} \)
89 \( 1 + (1.37 + 8.40i)T + (-84.3 + 28.4i)T^{2} \)
97 \( 1 + (0.182 - 3.36i)T + (-96.4 - 10.4i)T^{2} \)
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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

−12.45294068213292561498814811316, −11.93915561216407465716789008614, −10.80357131593125452950688314755, −10.14302298757279449517781208214, −8.449001391060186704181044635655, −7.87467831189411483767772498186, −6.61490322557756232433377813564, −5.27432069422105231962189141265, −4.16848702241746406053558789517, −2.25644722206121868611835257988, 1.62180381281329256183900142722, 3.02134532167508387752325631278, 5.10844782151808828931287517517, 6.16137193665147978776157094284, 7.52023226520766446293965137477, 8.085569179732630731158572137647, 9.795019469127428761240377867792, 10.78675868267059586586554483945, 11.62184244292262951795694548794, 12.16028879547992388330463915531

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