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Properties

Label 2-185-185.84-c1-0-5
Degree $2$
Conductor $185$
Sign $-0.136 - 0.990i$
Analytic cond. $1.47723$
Root an. cond. $1.21541$
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  + (1.64 + 0.949i)2-s + (−1.16 + 0.672i)3-s + (0.802 + 1.38i)4-s + (0.519 + 2.17i)5-s − 2.55·6-s + (−2.34 + 1.35i)7-s − 0.750i·8-s + (−0.596 + 1.03i)9-s + (−1.21 + 4.06i)10-s + 5.31·11-s + (−1.86 − 1.07i)12-s + (3.58 − 2.06i)13-s − 5.13·14-s + (−2.06 − 2.18i)15-s + (2.31 − 4.01i)16-s + (−0.0183 − 0.0106i)17-s + ⋯
L(s)  = 1  + (1.16 + 0.671i)2-s + (−0.672 + 0.388i)3-s + (0.401 + 0.694i)4-s + (0.232 + 0.972i)5-s − 1.04·6-s + (−0.885 + 0.511i)7-s − 0.265i·8-s + (−0.198 + 0.344i)9-s + (−0.382 + 1.28i)10-s + 1.60·11-s + (−0.539 − 0.311i)12-s + (0.994 − 0.574i)13-s − 1.37·14-s + (−0.533 − 0.563i)15-s + (0.579 − 1.00i)16-s + (−0.00445 − 0.00257i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.136 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.136 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.136 - 0.990i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (84, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ -0.136 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09359 + 1.25398i\)
\(L(\frac12)\) \(\approx\) \(1.09359 + 1.25398i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.519 - 2.17i)T \)
37 \( 1 + (-5.90 - 1.47i)T \)
good2 \( 1 + (-1.64 - 0.949i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (1.16 - 0.672i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (2.34 - 1.35i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 5.31T + 11T^{2} \)
13 \( 1 + (-3.58 + 2.06i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.0183 + 0.0106i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.01 + 6.96i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.35iT - 23T^{2} \)
29 \( 1 - 3.49T + 29T^{2} \)
31 \( 1 + 2.55T + 31T^{2} \)
41 \( 1 + (1.12 + 1.94i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 0.815iT - 43T^{2} \)
47 \( 1 + 12.2iT - 47T^{2} \)
53 \( 1 + (-0.432 - 0.249i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.95 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.05 - 3.56i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.39 - 5.42i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.71 + 9.90i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 10.6iT - 73T^{2} \)
79 \( 1 + (3.34 + 5.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.20 + 0.695i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.04 + 1.80i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.44iT - 97T^{2} \)
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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

−13.23398463447775702752186704459, −11.92423595540064525777698662412, −11.13687458520332218426752415978, −10.05287568276174253793412867287, −8.942716788776672048283024215858, −7.06907416102833867866271870455, −6.26211061449089631736875526993, −5.70882399243758727544149558425, −4.24911104721537544004521950961, −3.08750954782583872986749668353, 1.42061769457507082677704331557, 3.65058313440568580892732572107, 4.40793237666925459413653980293, 6.11592433825647745002279832869, 6.33520415800276622940060185572, 8.440616753804302191466450738480, 9.445116980739141642482479871731, 10.80679819011403002776552046879, 11.78888790860033942964683002966, 12.45046207812646691345337127008

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