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Properties

Label 2-185-185.43-c1-0-5
Degree $2$
Conductor $185$
Sign $-0.529 - 0.848i$
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  + 2.17i·2-s + (0.315 − 0.315i)3-s − 2.70·4-s + (2.17 − 0.539i)5-s + (0.684 + 0.684i)6-s + (−1.31 + 1.31i)7-s − 1.53i·8-s + 2.80i·9-s + (1.17 + 4.70i)10-s + 4.04i·11-s + (−0.854 + 0.854i)12-s − 5.80i·13-s + (−2.85 − 2.85i)14-s + (0.514 − 0.854i)15-s − 2.07·16-s + 2.87·17-s + ⋯
L(s)  = 1  + 1.53i·2-s + (0.182 − 0.182i)3-s − 1.35·4-s + (0.970 − 0.241i)5-s + (0.279 + 0.279i)6-s + (−0.497 + 0.497i)7-s − 0.544i·8-s + 0.933i·9-s + (0.370 + 1.48i)10-s + 1.22i·11-s + (−0.246 + 0.246i)12-s − 1.60i·13-s + (−0.762 − 0.762i)14-s + (0.132 − 0.220i)15-s − 0.519·16-s + 0.698·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 - 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.529 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.529 - 0.848i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ -0.529 - 0.848i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.630289 + 1.13630i\)
\(L(\frac12)\) \(\approx\) \(0.630289 + 1.13630i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.17 + 0.539i)T \)
37 \( 1 + (1 + 6i)T \)
good2 \( 1 - 2.17iT - 2T^{2} \)
3 \( 1 + (-0.315 + 0.315i)T - 3iT^{2} \)
7 \( 1 + (1.31 - 1.31i)T - 7iT^{2} \)
11 \( 1 - 4.04iT - 11T^{2} \)
13 \( 1 + 5.80iT - 13T^{2} \)
17 \( 1 - 2.87T + 17T^{2} \)
19 \( 1 + (-1.68 + 1.68i)T - 19iT^{2} \)
23 \( 1 + 8.04iT - 23T^{2} \)
29 \( 1 + (-0.0783 - 0.0783i)T + 29iT^{2} \)
31 \( 1 + (4.10 - 4.10i)T - 31iT^{2} \)
41 \( 1 + 5.46iT - 41T^{2} \)
43 \( 1 - 8.38iT - 43T^{2} \)
47 \( 1 + (-0.933 + 0.933i)T - 47iT^{2} \)
53 \( 1 + (4.17 + 4.17i)T + 53iT^{2} \)
59 \( 1 + (-8.10 + 8.10i)T - 59iT^{2} \)
61 \( 1 + (-6.70 + 6.70i)T - 61iT^{2} \)
67 \( 1 + (0.116 + 0.116i)T + 67iT^{2} \)
71 \( 1 + 7.75T + 71T^{2} \)
73 \( 1 + (7.09 - 7.09i)T - 73iT^{2} \)
79 \( 1 + (1.51 - 1.51i)T - 79iT^{2} \)
83 \( 1 + (-5.48 - 5.48i)T + 83iT^{2} \)
89 \( 1 + (2.61 + 2.61i)T + 89iT^{2} \)
97 \( 1 - 0.630T + 97T^{2} \)
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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

−12.95471704579428273076075687097, −12.65208506358739149390452538395, −10.61727633821123637473782987235, −9.694736358108892092192983583691, −8.628421682149726721781615092526, −7.71480540571919305689899282804, −6.72783829437329737250072071492, −5.57682970148646860977499302710, −4.95813127193799505888560496932, −2.48735242295633432778536816304, 1.42020209741084680288268981142, 3.10600102340992821212737598535, 3.91584371697693271680763101391, 5.78912548711155256315166067240, 6.94993160638223479331363503573, 8.934931289480042955305727670731, 9.569929028203685468498025712504, 10.23284272333570058452870499503, 11.39311730481726966728896654531, 11.97756255214975613687958695872

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