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Properties

Label 2-185-185.104-c1-0-6
Degree $2$
Conductor $185$
Sign $0.290 - 0.956i$
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  + (1.61 − 0.586i)2-s + (−0.898 + 2.46i)3-s + (0.722 − 0.606i)4-s + (−1.76 + 1.37i)5-s + 4.50i·6-s + (1.64 − 0.290i)7-s + (−0.906 + 1.57i)8-s + (−2.98 − 2.50i)9-s + (−2.03 + 3.25i)10-s + (−0.213 + 0.370i)11-s + (0.846 + 2.32i)12-s + (5.35 − 4.49i)13-s + (2.48 − 1.43i)14-s + (−1.81 − 5.58i)15-s + (−0.867 + 4.92i)16-s + (2.29 + 1.92i)17-s + ⋯
L(s)  = 1  + (1.13 − 0.414i)2-s + (−0.518 + 1.42i)3-s + (0.361 − 0.303i)4-s + (−0.787 + 0.616i)5-s + 1.83i·6-s + (0.621 − 0.109i)7-s + (−0.320 + 0.555i)8-s + (−0.995 − 0.835i)9-s + (−0.642 + 1.02i)10-s + (−0.0644 + 0.111i)11-s + (0.244 + 0.671i)12-s + (1.48 − 1.24i)13-s + (0.663 − 0.382i)14-s + (−0.469 − 1.44i)15-s + (−0.216 + 1.23i)16-s + (0.557 + 0.468i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.290 - 0.956i$
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: \((2,\ 185,\ (\ :1/2),\ 0.290 - 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25727 + 0.932091i\)
\(L(\frac12)\) \(\approx\) \(1.25727 + 0.932091i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.76 - 1.37i)T \)
37 \( 1 + (6.06 + 0.472i)T \)
good2 \( 1 + (-1.61 + 0.586i)T + (1.53 - 1.28i)T^{2} \)
3 \( 1 + (0.898 - 2.46i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (-1.64 + 0.290i)T + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (0.213 - 0.370i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.35 + 4.49i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-2.29 - 1.92i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (-1.68 + 4.61i)T + (-14.5 - 12.2i)T^{2} \)
23 \( 1 + (-3.21 - 5.56i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.17 + 3.56i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.0147iT - 31T^{2} \)
41 \( 1 + (-6.20 + 5.20i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 - 0.372T + 43T^{2} \)
47 \( 1 + (0.489 - 0.282i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.48 + 0.261i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-8.78 - 1.54i)T + (55.4 + 20.1i)T^{2} \)
61 \( 1 + (4.71 + 5.61i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (5.36 - 0.945i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-0.117 - 0.0426i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 - 9.60iT - 73T^{2} \)
79 \( 1 + (-3.72 + 0.655i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (-10.6 + 12.6i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (0.556 + 0.0981i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (-0.805 - 1.39i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.77540391131196701028007515938, −11.41740779516064671109161643602, −11.19318388184942236531543722807, −10.37960320479385304203091757535, −8.911929157239469759827347322004, −7.71461323423237519305947452270, −5.85187101832786912347541591499, −5.04867192911326456901671919682, −3.89498469516213194287493443022, −3.28865009149529507645458455540, 1.29186037913886708541068204702, 3.73963111323566859613972764743, 5.00515719208021326054758058468, 6.03100076633064295493613472715, 6.96990539569636003581864398768, 7.961463775771032234942990284184, 9.043209120742326277399316463814, 11.11375135280609348648038822140, 11.86717752980561734544379436538, 12.51148103638386709647218099647

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