Properties

Label 2-177-177.104-c2-0-18
Degree $2$
Conductor $177$
Sign $-0.733 - 0.679i$
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  + (1.88 + 3.13i)2-s + (2.92 + 0.677i)3-s + (−4.37 + 8.25i)4-s + (−0.175 − 1.61i)5-s + (3.38 + 10.4i)6-s + (0.807 − 0.271i)7-s + (−19.5 + 1.05i)8-s + (8.08 + 3.95i)9-s + (4.72 − 3.59i)10-s + (0.113 + 0.0769i)11-s + (−18.3 + 21.1i)12-s + (−4.49 − 4.25i)13-s + (2.37 + 2.01i)14-s + (0.580 − 4.84i)15-s + (−19.0 − 28.1i)16-s + (−2.77 + 8.23i)17-s + ⋯
L(s)  = 1  + (0.941 + 1.56i)2-s + (0.974 + 0.225i)3-s + (−1.09 + 2.06i)4-s + (−0.0351 − 0.323i)5-s + (0.564 + 1.73i)6-s + (0.115 − 0.0388i)7-s + (−2.43 + 0.132i)8-s + (0.898 + 0.439i)9-s + (0.472 − 0.359i)10-s + (0.0103 + 0.00699i)11-s + (−1.53 + 1.76i)12-s + (−0.345 − 0.327i)13-s + (0.169 + 0.143i)14-s + (0.0387 − 0.322i)15-s + (−1.19 − 1.75i)16-s + (−0.163 + 0.484i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.733 - 0.679i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.733 - 0.679i$
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: \((2,\ 177,\ (\ :1),\ -0.733 - 0.679i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.07552 + 2.74457i\)
\(L(\frac12)\) \(\approx\) \(1.07552 + 2.74457i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.92 - 0.677i)T \)
59 \( 1 + (-11.2 + 57.9i)T \)
good2 \( 1 + (-1.88 - 3.13i)T + (-1.87 + 3.53i)T^{2} \)
5 \( 1 + (0.175 + 1.61i)T + (-24.4 + 5.37i)T^{2} \)
7 \( 1 + (-0.807 + 0.271i)T + (39.0 - 29.6i)T^{2} \)
11 \( 1 + (-0.113 - 0.0769i)T + (44.7 + 112. i)T^{2} \)
13 \( 1 + (4.49 + 4.25i)T + (9.14 + 168. i)T^{2} \)
17 \( 1 + (2.77 - 8.23i)T + (-230. - 174. i)T^{2} \)
19 \( 1 + (5.30 + 32.3i)T + (-342. + 115. i)T^{2} \)
23 \( 1 + (-21.1 + 5.86i)T + (453. - 272. i)T^{2} \)
29 \( 1 + (3.98 - 6.61i)T + (-393. - 743. i)T^{2} \)
31 \( 1 + (3.27 - 19.9i)T + (-910. - 306. i)T^{2} \)
37 \( 1 + (-3.29 + 60.7i)T + (-1.36e3 - 148. i)T^{2} \)
41 \( 1 + (-3.82 - 1.06i)T + (1.44e3 + 866. i)T^{2} \)
43 \( 1 + (32.6 + 48.1i)T + (-684. + 1.71e3i)T^{2} \)
47 \( 1 + (9.16 - 84.2i)T + (-2.15e3 - 474. i)T^{2} \)
53 \( 1 + (32.1 - 42.2i)T + (-751. - 2.70e3i)T^{2} \)
61 \( 1 + (-20.4 + 12.3i)T + (1.74e3 - 3.28e3i)T^{2} \)
67 \( 1 + (-4.15 - 76.6i)T + (-4.46e3 + 485. i)T^{2} \)
71 \( 1 + (3.76 - 34.5i)T + (-4.92e3 - 1.08e3i)T^{2} \)
73 \( 1 + (-46.3 + 54.5i)T + (-862. - 5.25e3i)T^{2} \)
79 \( 1 + (7.03 - 17.6i)T + (-4.53e3 - 4.29e3i)T^{2} \)
83 \( 1 + (65.3 + 141. i)T + (-4.45e3 + 5.25e3i)T^{2} \)
89 \( 1 + (13.0 - 21.7i)T + (-3.71e3 - 6.99e3i)T^{2} \)
97 \( 1 + (71.8 + 84.5i)T + (-1.52e3 + 9.28e3i)T^{2} \)
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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

−12.95001063458010750593711495297, −12.72831055162572096529646505144, −10.86199874519486844377710275332, −9.183731815306178071158252149747, −8.570926542097139818117863353050, −7.50312544950881687238914948732, −6.69849474393926342230951250077, −5.13998082331907442210151468615, −4.37053100049658858843472806085, −2.98442602733381768930288166008, 1.57611583746690988724999694213, 2.82751429122834439997500638733, 3.81187568833603327651429541832, 5.03488413384263555634385336728, 6.67255750959901531046157777708, 8.220426394380465333619837529745, 9.506762671747739748197726713321, 10.16574290269712093092450400740, 11.31866385500816094345022858864, 12.17106151828866705826103343646

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