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Properties

Label 2-185-185.84-c1-0-4
Degree $2$
Conductor $185$
Sign $-0.613 - 0.789i$
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  + (2.33 + 1.34i)2-s + (−2.35 + 1.36i)3-s + (2.63 + 4.57i)4-s + (−2.05 − 0.884i)5-s − 7.33·6-s + (1.64 − 0.948i)7-s + 8.84i·8-s + (2.20 − 3.81i)9-s + (−3.60 − 4.83i)10-s + 3.17·11-s + (−12.4 − 7.18i)12-s + (−2.13 + 1.23i)13-s + 5.12·14-s + (6.04 − 0.710i)15-s + (−6.65 + 11.5i)16-s + (4.72 + 2.72i)17-s + ⋯
L(s)  = 1  + (1.65 + 0.953i)2-s + (−1.36 + 0.785i)3-s + (1.31 + 2.28i)4-s + (−0.918 − 0.395i)5-s − 2.99·6-s + (0.621 − 0.358i)7-s + 3.12i·8-s + (0.733 − 1.27i)9-s + (−1.14 − 1.52i)10-s + 0.958·11-s + (−3.59 − 2.07i)12-s + (−0.592 + 0.342i)13-s + 1.36·14-s + (1.55 − 0.183i)15-s + (−1.66 + 2.88i)16-s + (1.14 + 0.661i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 - 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.613 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.613 - 0.789i$
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: \((2,\ 185,\ (\ :1/2),\ -0.613 - 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.780677 + 1.59604i\)
\(L(\frac12)\) \(\approx\) \(0.780677 + 1.59604i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.05 + 0.884i)T \)
37 \( 1 + (4.84 + 3.68i)T \)
good2 \( 1 + (-2.33 - 1.34i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (2.35 - 1.36i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-1.64 + 0.948i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 3.17T + 11T^{2} \)
13 \( 1 + (2.13 - 1.23i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-4.72 - 2.72i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.821 + 1.42i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.20iT - 23T^{2} \)
29 \( 1 - 0.245T + 29T^{2} \)
31 \( 1 - 1.81T + 31T^{2} \)
41 \( 1 + (2.88 + 4.98i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 1.43iT - 43T^{2} \)
47 \( 1 + 4.32iT - 47T^{2} \)
53 \( 1 + (6.15 + 3.55i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.57 + 6.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.97 - 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.73 - 5.62i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.12 + 5.41i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 0.285iT - 73T^{2} \)
79 \( 1 + (-1.36 - 2.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.71 + 2.14i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.885 - 1.53i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.52iT - 97T^{2} \)
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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

−12.71010995765317311819889204789, −12.03537390942847074882398247822, −11.54699193745887582072564712692, −10.50577055729393922803301432493, −8.546969080555225648651826288685, −7.32567693769024043313821460217, −6.38002176733527382459371166228, −5.23219751567714048864203972921, −4.49969744639048868211052008779, −3.78503285938260771126473482956, 1.39775406393756918928857514364, 3.28966295977939389828994922351, 4.71445738017289922749499639315, 5.56310024494911159185586875249, 6.59147673339102139960301578045, 7.58618716230554711163563286029, 9.956176233577095470289985719097, 11.06609755004058476039624624182, 11.84004074470016085268506619177, 11.88441649812152506218110725757

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