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Properties

Label 2-185-185.84-c1-0-2
Degree $2$
Conductor $185$
Sign $0.992 - 0.125i$
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  + (−1.58 − 0.914i)2-s + (−1.14 + 0.663i)3-s + (0.673 + 1.16i)4-s + (2.12 − 0.686i)5-s + 2.42·6-s + (−2.49 + 1.44i)7-s + 1.19i·8-s + (−0.618 + 1.07i)9-s + (−4.00 − 0.858i)10-s + 1.49·11-s + (−1.54 − 0.894i)12-s + (5.01 − 2.89i)13-s + 5.27·14-s + (−1.99 + 2.20i)15-s + (2.43 − 4.22i)16-s + (5.81 + 3.35i)17-s + ⋯
L(s)  = 1  + (−1.12 − 0.646i)2-s + (−0.663 + 0.383i)3-s + (0.336 + 0.583i)4-s + (0.951 − 0.307i)5-s + 0.991·6-s + (−0.944 + 0.545i)7-s + 0.421i·8-s + (−0.206 + 0.357i)9-s + (−1.26 − 0.271i)10-s + 0.450·11-s + (−0.447 − 0.258i)12-s + (1.39 − 0.802i)13-s + 1.41·14-s + (−0.514 + 0.568i)15-s + (0.609 − 1.05i)16-s + (1.41 + 0.814i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.992 - 0.125i$
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: \((2,\ 185,\ (\ :1/2),\ 0.992 - 0.125i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.580815 + 0.0367124i\)
\(L(\frac12)\) \(\approx\) \(0.580815 + 0.0367124i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.12 + 0.686i)T \)
37 \( 1 + (5.74 + 1.98i)T \)
good2 \( 1 + (1.58 + 0.914i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (1.14 - 0.663i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (2.49 - 1.44i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 1.49T + 11T^{2} \)
13 \( 1 + (-5.01 + 2.89i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-5.81 - 3.35i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.90 - 3.30i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.31iT - 23T^{2} \)
29 \( 1 + 6.15T + 29T^{2} \)
31 \( 1 - 2.38T + 31T^{2} \)
41 \( 1 + (-3.78 - 6.55i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 6.09iT - 43T^{2} \)
47 \( 1 - 7.76iT - 47T^{2} \)
53 \( 1 + (-0.317 - 0.183i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.43 + 7.68i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.97 - 6.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.10 - 4.10i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.44 - 2.49i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.1iT - 73T^{2} \)
79 \( 1 + (4.49 + 7.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.13 + 4.11i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.35 - 4.07i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.47iT - 97T^{2} \)
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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

−12.35528714054231581478218926527, −11.34749853071422848729003427800, −10.38346611444806406230313722869, −9.836601696434232405730123684641, −8.965298662307420000074167533629, −7.942929104361670553087508407667, −5.79847726512021471704512466222, −5.69057774222535554432472370239, −3.29379142291040841990774667910, −1.45822675803674556064186983727, 0.964942342305679896606127302598, 3.50946029946590753269226509897, 5.75749312062990398938131085605, 6.62317791701129826614930665995, 7.08725476506518820156996745811, 8.759654364246315548519671360620, 9.472485115940926343227377986822, 10.30398441810819388469112913577, 11.41104862245379043523293331174, 12.62464631431126107626969091012

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