↑

Properties

Label 2-185-185.84-c1-0-9
Degree $2$
Conductor $185$
Sign $-0.925 - 0.379i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.99 − 1.15i)2-s + (−1.81 + 1.04i)3-s + (1.66 + 2.88i)4-s + (−0.0902 − 2.23i)5-s + 4.82·6-s + (3.28 − 1.89i)7-s − 3.06i·8-s + (0.687 − 1.18i)9-s + (−2.39 + 4.56i)10-s − 3.98·11-s + (−6.02 − 3.47i)12-s + (−5.14 + 2.96i)13-s − 8.75·14-s + (2.49 + 3.95i)15-s + (−0.206 + 0.358i)16-s + (−2.60 − 1.50i)17-s + ⋯
L(s)  = 1  + (−1.41 − 0.815i)2-s + (−1.04 + 0.603i)3-s + (0.831 + 1.44i)4-s + (−0.0403 − 0.999i)5-s + 1.97·6-s + (1.24 − 0.717i)7-s − 1.08i·8-s + (0.229 − 0.396i)9-s + (−0.758 + 1.44i)10-s − 1.20·11-s + (−1.73 − 1.00i)12-s + (−1.42 + 0.823i)13-s − 2.34·14-s + (0.645 + 1.02i)15-s + (−0.0516 + 0.0895i)16-s + (−0.630 − 0.364i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 - 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.925 - 0.379i$
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: \((2,\ 185,\ (\ :1/2),\ -0.925 - 0.379i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0189202 + 0.0960173i\)
\(L(\frac12)\) \(\approx\) \(0.0189202 + 0.0960173i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.0902 + 2.23i)T \)
37 \( 1 + (3.31 - 5.09i)T \)
good2 \( 1 + (1.99 + 1.15i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (1.81 - 1.04i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-3.28 + 1.89i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 3.98T + 11T^{2} \)
13 \( 1 + (5.14 - 2.96i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.60 + 1.50i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.477 + 0.826i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.297iT - 23T^{2} \)
29 \( 1 + 6.45T + 29T^{2} \)
31 \( 1 + 1.69T + 31T^{2} \)
41 \( 1 + (-3.89 - 6.74i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 4.65iT - 43T^{2} \)
47 \( 1 + 3.36iT - 47T^{2} \)
53 \( 1 + (-1.67 - 0.965i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.90 + 5.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.95 + 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.15 - 4.70i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.41 + 11.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.763iT - 73T^{2} \)
79 \( 1 + (-4.90 - 8.50i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-13.4 - 7.77i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.32 - 4.02i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.16iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.55433448038957787155780440078, −11.03092115704888914931363517376, −10.17078209270571940732546508208, −9.322193483257754666345306738902, −8.171653630329938685386112447756, −7.37551407577747688903554346748, −5.18553255913220587327753406010, −4.58003275341020632875458449328, −2.00386040494268689979456981974, −0.14674645848734607205827683679, 2.18213506927382937833881523944, 5.29898112266868999346294843567, 6.02795013618244264725647585661, 7.43970613334812770901568223723, 7.62582726501060939046596004499, 8.976761627930307706188875995175, 10.38099256165158647987117457544, 10.86837888783062607298505792268, 11.85817495587716865677794700536, 12.92896801792134640522083662760

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