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Properties

Label 2-185-185.4-c1-0-2
Degree $2$
Conductor $185$
Sign $-0.999 - 0.0446i$
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  + (−0.430 + 2.43i)2-s + (1.12 − 0.198i)3-s + (−3.88 − 1.41i)4-s + (−1.62 + 1.54i)5-s + 2.83i·6-s + (1.92 + 2.28i)7-s + (2.64 − 4.58i)8-s + (−1.58 + 0.577i)9-s + (−3.06 − 4.61i)10-s + (−0.267 + 0.463i)11-s + (−4.66 − 0.822i)12-s + (−2.41 − 0.877i)13-s + (−6.41 + 3.70i)14-s + (−1.52 + 2.06i)15-s + (3.70 + 3.11i)16-s + (1.07 − 0.391i)17-s + ⋯
L(s)  = 1  + (−0.304 + 1.72i)2-s + (0.651 − 0.114i)3-s + (−1.94 − 0.707i)4-s + (−0.724 + 0.689i)5-s + 1.15i·6-s + (0.726 + 0.865i)7-s + (0.935 − 1.62i)8-s + (−0.528 + 0.192i)9-s + (−0.968 − 1.45i)10-s + (−0.0807 + 0.139i)11-s + (−1.34 − 0.237i)12-s + (−0.668 − 0.243i)13-s + (−1.71 + 0.989i)14-s + (−0.392 + 0.532i)15-s + (0.927 + 0.778i)16-s + (0.260 − 0.0949i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.999 - 0.0446i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ -0.999 - 0.0446i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0213405 + 0.955481i\)
\(L(\frac12)\) \(\approx\) \(0.0213405 + 0.955481i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.62 - 1.54i)T \)
37 \( 1 + (5.66 + 2.22i)T \)
good2 \( 1 + (0.430 - 2.43i)T + (-1.87 - 0.684i)T^{2} \)
3 \( 1 + (-1.12 + 0.198i)T + (2.81 - 1.02i)T^{2} \)
7 \( 1 + (-1.92 - 2.28i)T + (-1.21 + 6.89i)T^{2} \)
11 \( 1 + (0.267 - 0.463i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.41 + 0.877i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-1.07 + 0.391i)T + (13.0 - 10.9i)T^{2} \)
19 \( 1 + (-6.95 + 1.22i)T + (17.8 - 6.49i)T^{2} \)
23 \( 1 + (-3.82 - 6.63i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.54 - 2.62i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.99iT - 31T^{2} \)
41 \( 1 + (-6.61 - 2.40i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 - 10.0T + 43T^{2} \)
47 \( 1 + (1.01 - 0.583i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.02 - 3.60i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (3.09 - 3.68i)T + (-10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.833 - 2.28i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (8.36 + 9.96i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (1.13 + 6.42i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + 10.5iT - 73T^{2} \)
79 \( 1 + (-3.98 - 4.74i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (1.74 + 4.78i)T + (-63.5 + 53.3i)T^{2} \)
89 \( 1 + (5.26 - 6.27i)T + (-15.4 - 87.6i)T^{2} \)
97 \( 1 + (-2.38 - 4.12i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.78048714103746187241546260755, −12.15305625504191737177665539641, −11.13248029354754772254458316313, −9.513899237232675794569417306108, −8.740814842302265439375729846772, −7.57845672051770359084722323133, −7.49856629672456475037590260271, −5.81669055971202859926035629192, −4.92063261514159707248001091199, −2.98629462245716808328679064661, 0.975427404685186817409628520456, 2.82085177002613139516076329705, 3.95757149631505552653001154128, 4.93685088183867288231839437693, 7.54061846741374801521636078149, 8.446166365293305780778128138851, 9.222456815338841155502910971742, 10.26415986183038585989064296631, 11.24474585447403621902814987987, 11.95454464465530651202396718121

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