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Properties

Label 2-185-185.84-c1-0-17
Degree $2$
Conductor $185$
Sign $-0.939 - 0.342i$
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  + (−0.704 − 0.406i)2-s + (−0.142 + 0.0823i)3-s + (−0.668 − 1.15i)4-s + (−1.50 − 1.65i)5-s + 0.134·6-s + (−2.76 + 1.59i)7-s + 2.71i·8-s + (−1.48 + 2.57i)9-s + (0.386 + 1.77i)10-s + 0.642·11-s + (0.190 + 0.110i)12-s + (0.0879 − 0.0507i)13-s + 2.60·14-s + (0.350 + 0.112i)15-s + (−0.232 + 0.403i)16-s + (−5.39 − 3.11i)17-s + ⋯
L(s)  = 1  + (−0.498 − 0.287i)2-s + (−0.0823 + 0.0475i)3-s + (−0.334 − 0.579i)4-s + (−0.672 − 0.740i)5-s + 0.0547·6-s + (−1.04 + 0.604i)7-s + 0.960i·8-s + (−0.495 + 0.858i)9-s + (0.122 + 0.562i)10-s + 0.193·11-s + (0.0551 + 0.0318i)12-s + (0.0243 − 0.0140i)13-s + 0.695·14-s + (0.0906 + 0.0289i)15-s + (−0.0581 + 0.100i)16-s + (−1.30 − 0.755i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.939 - 0.342i$
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: \((2,\ 185,\ (\ :1/2),\ -0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0195421 + 0.110803i\)
\(L(\frac12)\) \(\approx\) \(0.0195421 + 0.110803i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.50 + 1.65i)T \)
37 \( 1 + (5.74 - 2.01i)T \)
good2 \( 1 + (0.704 + 0.406i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.142 - 0.0823i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (2.76 - 1.59i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 0.642T + 11T^{2} \)
13 \( 1 + (-0.0879 + 0.0507i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.39 + 3.11i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.58 + 2.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.40iT - 23T^{2} \)
29 \( 1 + 0.922T + 29T^{2} \)
31 \( 1 - 2.12T + 31T^{2} \)
41 \( 1 + (3.24 + 5.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 4.73iT - 43T^{2} \)
47 \( 1 + 11.7iT - 47T^{2} \)
53 \( 1 + (-4.38 - 2.53i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.782 - 1.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.43 - 7.68i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.52 - 0.882i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.31 - 10.9i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 5.29iT - 73T^{2} \)
79 \( 1 + (4.11 + 7.11i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.85 + 5.68i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.258 - 0.448i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.2iT - 97T^{2} \)
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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

−11.89355489735735773702182546786, −11.05025288800669806837735952562, −10.02492228088738233263424121776, −8.821494233680301446637574262249, −8.593283089704501548063420102925, −6.83636066799485206537966867474, −5.47714653136666807850998995529, −4.48780814720920609216013954313, −2.48130546325236168644004783672, −0.11479275981471446584249419458, 3.32949593738461134778266842867, 4.00179631950007657611967289804, 6.33422376960800423492352166052, 6.94921794553331742241502436181, 8.067171431732658431991755268705, 9.098104038732898696451600745714, 10.03424088579371276457886656463, 11.17123090167085765036797478514, 12.21094661320021419328307411295, 13.04824610790335024215504080298

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