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Properties

Label 2-185-185.104-c1-0-7
Degree $2$
Conductor $185$
Sign $0.985 - 0.171i$
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  + (−2.11 + 0.768i)2-s + (0.602 − 1.65i)3-s + (2.33 − 1.95i)4-s + (0.876 + 2.05i)5-s + 3.96i·6-s + (−1.05 + 0.185i)7-s + (−1.17 + 2.03i)8-s + (−0.0829 − 0.0696i)9-s + (−3.43 − 3.66i)10-s + (−0.107 + 0.186i)11-s + (−1.83 − 5.04i)12-s + (4.54 − 3.81i)13-s + (2.08 − 1.20i)14-s + (3.93 − 0.211i)15-s + (−0.140 + 0.798i)16-s + (3.34 + 2.80i)17-s + ⋯
L(s)  = 1  + (−1.49 + 0.543i)2-s + (0.348 − 0.956i)3-s + (1.16 − 0.979i)4-s + (0.392 + 0.919i)5-s + 1.61i·6-s + (−0.398 + 0.0701i)7-s + (−0.415 + 0.720i)8-s + (−0.0276 − 0.0232i)9-s + (−1.08 − 1.16i)10-s + (−0.0324 + 0.0562i)11-s + (−0.530 − 1.45i)12-s + (1.26 − 1.05i)13-s + (0.556 − 0.321i)14-s + (1.01 − 0.0547i)15-s + (−0.0351 + 0.199i)16-s + (0.811 + 0.680i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.985 - 0.171i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ 0.985 - 0.171i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.715006 + 0.0616372i\)
\(L(\frac12)\) \(\approx\) \(0.715006 + 0.0616372i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.876 - 2.05i)T \)
37 \( 1 + (-4.84 + 3.68i)T \)
good2 \( 1 + (2.11 - 0.768i)T + (1.53 - 1.28i)T^{2} \)
3 \( 1 + (-0.602 + 1.65i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (1.05 - 0.185i)T + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (0.107 - 0.186i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.54 + 3.81i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-3.34 - 2.80i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (-1.34 + 3.69i)T + (-14.5 - 12.2i)T^{2} \)
23 \( 1 + (-1.58 - 2.75i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.22 - 1.86i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.43iT - 31T^{2} \)
41 \( 1 + (-1.58 + 1.32i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + 9.40T + 43T^{2} \)
47 \( 1 + (9.33 - 5.39i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.14 + 1.08i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (2.70 + 0.476i)T + (55.4 + 20.1i)T^{2} \)
61 \( 1 + (-5.71 - 6.81i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (9.05 - 1.59i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (1.81 + 0.659i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 - 0.242iT - 73T^{2} \)
79 \( 1 + (-1.58 + 0.279i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (0.918 - 1.09i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (9.15 + 1.61i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (-8.07 - 13.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.88923846366484319548701002868, −11.23296478861144206173100477662, −10.39996902353637487555014955484, −9.561566026202511278761921005747, −8.337913663510587671574705692309, −7.65284632472491939953935168818, −6.73585015914641623907115940432, −5.93962278718070573882048059498, −3.09040316449969749730364247451, −1.39910787551430922511705114938, 1.37945145582683974796695279843, 3.34265005249370937635285080407, 4.79578380824275083992298735256, 6.50810752415490660923970287820, 8.144723169616343732997796847272, 8.844291722869353430061445435316, 9.695105576688547899977624933811, 10.06606583565718049713058039499, 11.30236621315318858596339307283, 12.21682745769393711202357015873

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