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Properties

Label 2-185-185.104-c1-0-15
Degree $2$
Conductor $185$
Sign $0.451 + 0.892i$
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  + (1.68 − 0.611i)2-s + (0.595 − 1.63i)3-s + (0.919 − 0.771i)4-s + (2.09 − 0.793i)5-s − 3.11i·6-s + (−4.26 + 0.752i)7-s + (−0.714 + 1.23i)8-s + (−0.0222 − 0.0186i)9-s + (3.02 − 2.61i)10-s + (−1.91 + 3.31i)11-s + (−0.714 − 1.96i)12-s + (1.38 − 1.16i)13-s + (−6.71 + 3.87i)14-s + (−0.0528 − 3.89i)15-s + (−0.861 + 4.88i)16-s + (2.55 + 2.14i)17-s + ⋯
L(s)  = 1  + (1.18 − 0.432i)2-s + (0.343 − 0.944i)3-s + (0.459 − 0.385i)4-s + (0.934 − 0.354i)5-s − 1.27i·6-s + (−1.61 + 0.284i)7-s + (−0.252 + 0.437i)8-s + (−0.00740 − 0.00621i)9-s + (0.957 − 0.826i)10-s + (−0.577 + 1.00i)11-s + (−0.206 − 0.566i)12-s + (0.383 − 0.322i)13-s + (−1.79 + 1.03i)14-s + (−0.0136 − 1.00i)15-s + (−0.215 + 1.22i)16-s + (0.620 + 0.520i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.451 + 0.892i$
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: \((2,\ 185,\ (\ :1/2),\ 0.451 + 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84119 - 1.13225i\)
\(L(\frac12)\) \(\approx\) \(1.84119 - 1.13225i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.09 + 0.793i)T \)
37 \( 1 + (3.35 - 5.07i)T \)
good2 \( 1 + (-1.68 + 0.611i)T + (1.53 - 1.28i)T^{2} \)
3 \( 1 + (-0.595 + 1.63i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (4.26 - 0.752i)T + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (1.91 - 3.31i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.38 + 1.16i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-2.55 - 2.14i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (-1.92 + 5.28i)T + (-14.5 - 12.2i)T^{2} \)
23 \( 1 + (2.83 + 4.90i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.78 + 3.33i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.65iT - 31T^{2} \)
41 \( 1 + (3.08 - 2.58i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + 4.20T + 43T^{2} \)
47 \( 1 + (0.200 - 0.116i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.06 - 1.42i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (4.92 + 0.868i)T + (55.4 + 20.1i)T^{2} \)
61 \( 1 + (8.08 + 9.63i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (4.20 - 0.741i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-10.8 - 3.96i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + 1.91iT - 73T^{2} \)
79 \( 1 + (-1.53 + 0.270i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (2.48 - 2.96i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (-14.5 - 2.57i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (-2.44 - 4.23i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.59459920866959374957741255392, −12.27112900847630835531245431115, −10.48157133334474870795832907143, −9.575554083311593530896399880077, −8.368723379278874514816273995193, −6.87657598432591468450579603778, −6.00841238320103246419621104042, −4.85044139940958903604454403796, −3.14877338704506158574203696133, −2.12735616082866345420139448089, 3.27309579239234185876298762184, 3.69127673350003551678930985477, 5.47518076944089825424705001081, 6.05367134106707346256760365250, 7.21503398305356972807206195484, 9.205989660536644901773167570244, 9.768072502076648390153123479894, 10.53861154263794006755812089267, 12.16378004878882865430171262121, 13.27953382741595388790145110669

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