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Properties

Label 2-185-185.103-c1-0-16
Degree $2$
Conductor $185$
Sign $-0.175 - 0.984i$
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  + (−1.29 − 2.23i)2-s + (−0.642 − 2.39i)3-s + (−2.33 + 4.05i)4-s + (−0.420 − 2.19i)5-s + (−4.53 + 4.53i)6-s + (−0.287 − 1.07i)7-s + 6.92·8-s + (−2.73 + 1.57i)9-s + (−4.37 + 3.77i)10-s − 1.13i·11-s + (11.2 + 3.00i)12-s + (−1.96 + 3.40i)13-s + (−2.03 + 2.03i)14-s + (−4.99 + 2.41i)15-s + (−4.26 − 7.39i)16-s + (6.66 − 3.84i)17-s + ⋯
L(s)  = 1  + (−0.913 − 1.58i)2-s + (−0.370 − 1.38i)3-s + (−1.16 + 2.02i)4-s + (−0.187 − 0.982i)5-s + (−1.85 + 1.85i)6-s + (−0.108 − 0.405i)7-s + 2.44·8-s + (−0.911 + 0.526i)9-s + (−1.38 + 1.19i)10-s − 0.342i·11-s + (3.23 + 0.867i)12-s + (−0.545 + 0.945i)13-s + (−0.542 + 0.542i)14-s + (−1.28 + 0.624i)15-s + (−1.06 − 1.84i)16-s + (1.61 − 0.933i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.175 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.175 - 0.984i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ -0.175 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.318541 + 0.380300i\)
\(L(\frac12)\) \(\approx\) \(0.318541 + 0.380300i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.420 + 2.19i)T \)
37 \( 1 + (6.08 + 0.0901i)T \)
good2 \( 1 + (1.29 + 2.23i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.642 + 2.39i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.287 + 1.07i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + 1.13iT - 11T^{2} \)
13 \( 1 + (1.96 - 3.40i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-6.66 + 3.84i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.44 + 1.72i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 2.90T + 23T^{2} \)
29 \( 1 + (5.98 - 5.98i)T - 29iT^{2} \)
31 \( 1 + (2.00 + 2.00i)T + 31iT^{2} \)
41 \( 1 + (-2.72 - 1.57i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + 3.89T + 43T^{2} \)
47 \( 1 + (-0.267 + 0.267i)T - 47iT^{2} \)
53 \( 1 + (-0.908 + 3.38i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.528 + 1.97i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (4.38 - 1.17i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-8.91 + 2.38i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-4.58 + 7.94i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.472 - 0.472i)T - 73iT^{2} \)
79 \( 1 + (-12.9 + 3.45i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-2.38 + 8.90i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (0.295 + 0.0791i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 - 15.1iT - 97T^{2} \)
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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

−11.99465376419113401228781854286, −11.28586734328353384603740966147, −9.826550120307540147449321833399, −9.133017329722872788004577195666, −7.86647621033462293507834025496, −7.25775943044817824365841720992, −5.24606591825710402937020248893, −3.50130969706443191272782716149, −1.75900880110663608815132860674, −0.66547694064704057880428981475, 3.64655390038250741474529048953, 5.36454797091230987640355878701, 5.83631530644373083483857062021, 7.34901095223680019362970316263, 8.059539352212466788099850840910, 9.578435868347424662765493261597, 9.959983114156390151787801552869, 10.71225455511132513539374951607, 12.12124540162630245612225763988, 14.03199063043419847903944882564

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