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Properties

Label 2-185-185.99-c1-0-10
Degree $2$
Conductor $185$
Sign $0.574 + 0.818i$
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  + (−2.05 + 1.72i)2-s + (1.45 − 1.73i)3-s + (0.897 − 5.09i)4-s + (−2.01 + 0.978i)5-s + 6.04i·6-s + (−0.461 − 1.26i)7-s + (4.24 + 7.35i)8-s + (−0.365 − 2.07i)9-s + (2.44 − 5.46i)10-s + (−2.25 − 3.91i)11-s + (−7.50 − 8.94i)12-s + (0.888 − 5.04i)13-s + (3.13 + 1.80i)14-s + (−1.22 + 4.90i)15-s + (−11.6 − 4.23i)16-s + (−0.248 − 1.41i)17-s + ⋯
L(s)  = 1  + (−1.45 + 1.21i)2-s + (0.838 − 0.999i)3-s + (0.448 − 2.54i)4-s + (−0.899 + 0.437i)5-s + 2.46i·6-s + (−0.174 − 0.479i)7-s + (1.50 + 2.59i)8-s + (−0.121 − 0.690i)9-s + (0.771 − 1.72i)10-s + (−0.680 − 1.17i)11-s + (−2.16 − 2.58i)12-s + (0.246 − 1.39i)13-s + (0.836 + 0.483i)14-s + (−0.316 + 1.26i)15-s + (−2.91 − 1.05i)16-s + (−0.0603 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.574 + 0.818i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ 0.574 + 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.482492 - 0.250729i\)
\(L(\frac12)\) \(\approx\) \(0.482492 - 0.250729i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.01 - 0.978i)T \)
37 \( 1 + (6.03 - 0.783i)T \)
good2 \( 1 + (2.05 - 1.72i)T + (0.347 - 1.96i)T^{2} \)
3 \( 1 + (-1.45 + 1.73i)T + (-0.520 - 2.95i)T^{2} \)
7 \( 1 + (0.461 + 1.26i)T + (-5.36 + 4.49i)T^{2} \)
11 \( 1 + (2.25 + 3.91i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.888 + 5.04i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (0.248 + 1.41i)T + (-15.9 + 5.81i)T^{2} \)
19 \( 1 + (-0.146 + 0.174i)T + (-3.29 - 18.7i)T^{2} \)
23 \( 1 + (-2.55 + 4.43i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.62 - 2.09i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.44iT - 31T^{2} \)
41 \( 1 + (-1.19 + 6.76i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 - 0.770T + 43T^{2} \)
47 \( 1 + (-8.65 - 4.99i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.51 - 4.14i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (-0.825 + 2.26i)T + (-45.1 - 37.9i)T^{2} \)
61 \( 1 + (2.61 + 0.460i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-3.62 - 9.95i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (-1.03 - 0.870i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 - 6.70iT - 73T^{2} \)
79 \( 1 + (-1.76 - 4.84i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (7.50 - 1.32i)T + (77.9 - 28.3i)T^{2} \)
89 \( 1 + (-0.458 + 1.26i)T + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (-7.82 + 13.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.64759189640875171996675176363, −10.86417889435571826159753635902, −10.47242106818027792391131900028, −8.726546230647186813282930439898, −8.330000369805469521513109975624, −7.42559459726302176907381275534, −6.90808409601433838045294572880, −5.52213881853292505355022019043, −2.97267044739247430365585738990, −0.71593054808011345389270996466, 2.14120317776186304114690771782, 3.55244339359673776981832161884, 4.43477231217115688008958523389, 7.29488237576775066438054823517, 8.184671796073466581311949230197, 9.229739857335185245684883314521, 9.436132371787754433562413326621, 10.58920010858035227834947963414, 11.57220769018340365006196137436, 12.29375033077554174299574663149

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