Properties

Label 2-177-177.104-c2-0-6
Degree $2$
Conductor $177$
Sign $0.881 - 0.472i$
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  + (−1.62 − 2.69i)2-s + (1.80 + 2.39i)3-s + (−2.75 + 5.20i)4-s + (0.0668 + 0.615i)5-s + (3.51 − 8.75i)6-s + (−2.85 + 0.963i)7-s + (5.92 − 0.321i)8-s + (−2.45 + 8.66i)9-s + (1.54 − 1.17i)10-s + (1.45 + 0.985i)11-s + (−17.4 + 2.81i)12-s + (8.35 + 7.91i)13-s + (7.23 + 6.14i)14-s + (−1.35 + 1.27i)15-s + (2.74 + 4.05i)16-s + (−7.09 + 21.0i)17-s + ⋯
L(s)  = 1  + (−0.810 − 1.34i)2-s + (0.603 + 0.797i)3-s + (−0.689 + 1.30i)4-s + (0.0133 + 0.123i)5-s + (0.585 − 1.45i)6-s + (−0.408 + 0.137i)7-s + (0.740 − 0.0401i)8-s + (−0.272 + 0.962i)9-s + (0.154 − 0.117i)10-s + (0.132 + 0.0895i)11-s + (−1.45 + 0.234i)12-s + (0.642 + 0.608i)13-s + (0.516 + 0.438i)14-s + (−0.0900 + 0.0848i)15-s + (0.171 + 0.253i)16-s + (−0.417 + 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.881 - 0.472i$
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: \((2,\ 177,\ (\ :1),\ 0.881 - 0.472i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.922168 + 0.231542i\)
\(L(\frac12)\) \(\approx\) \(0.922168 + 0.231542i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.80 - 2.39i)T \)
59 \( 1 + (-58.1 + 10.0i)T \)
good2 \( 1 + (1.62 + 2.69i)T + (-1.87 + 3.53i)T^{2} \)
5 \( 1 + (-0.0668 - 0.615i)T + (-24.4 + 5.37i)T^{2} \)
7 \( 1 + (2.85 - 0.963i)T + (39.0 - 29.6i)T^{2} \)
11 \( 1 + (-1.45 - 0.985i)T + (44.7 + 112. i)T^{2} \)
13 \( 1 + (-8.35 - 7.91i)T + (9.14 + 168. i)T^{2} \)
17 \( 1 + (7.09 - 21.0i)T + (-230. - 174. i)T^{2} \)
19 \( 1 + (-3.53 - 21.5i)T + (-342. + 115. i)T^{2} \)
23 \( 1 + (3.35 - 0.932i)T + (453. - 272. i)T^{2} \)
29 \( 1 + (-6.22 + 10.3i)T + (-393. - 743. i)T^{2} \)
31 \( 1 + (-3.23 + 19.7i)T + (-910. - 306. i)T^{2} \)
37 \( 1 + (-0.0142 + 0.262i)T + (-1.36e3 - 148. i)T^{2} \)
41 \( 1 + (29.9 + 8.31i)T + (1.44e3 + 866. i)T^{2} \)
43 \( 1 + (-6.65 - 9.81i)T + (-684. + 1.71e3i)T^{2} \)
47 \( 1 + (0.778 - 7.15i)T + (-2.15e3 - 474. i)T^{2} \)
53 \( 1 + (34.8 - 45.8i)T + (-751. - 2.70e3i)T^{2} \)
61 \( 1 + (-9.21 + 5.54i)T + (1.74e3 - 3.28e3i)T^{2} \)
67 \( 1 + (-0.638 - 11.7i)T + (-4.46e3 + 485. i)T^{2} \)
71 \( 1 + (-2.88 + 26.5i)T + (-4.92e3 - 1.08e3i)T^{2} \)
73 \( 1 + (-48.7 + 57.4i)T + (-862. - 5.25e3i)T^{2} \)
79 \( 1 + (-43.5 + 109. i)T + (-4.53e3 - 4.29e3i)T^{2} \)
83 \( 1 + (31.2 + 67.5i)T + (-4.45e3 + 5.25e3i)T^{2} \)
89 \( 1 + (83.2 - 138. i)T + (-3.71e3 - 6.99e3i)T^{2} \)
97 \( 1 + (63.9 + 75.3i)T + (-1.52e3 + 9.28e3i)T^{2} \)
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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

−12.29829678061179357871710194184, −11.20104686320435260490430468507, −10.45837619153200164891286879609, −9.692047721374281818259503179833, −8.831050719303809788478126169659, −8.060327517342948457322050362938, −6.15519742638416758776264033351, −4.19848649074757980422905382988, −3.24471316164432351822870661780, −1.85537505949465297590495948042, 0.71461531610745965217758419328, 3.06012498654117617832534601368, 5.24345840461280274440985568352, 6.64024023861517428072191384043, 7.07146780255650229948351354333, 8.289640438901282485778410713461, 8.921451208598684165514204928263, 9.832770732389767201331425201266, 11.36839496628433969115412470233, 12.69019739021587791316831692855

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