↑

Properties

Label 2-185-37.10-c1-0-5
Degree $2$
Conductor $185$
Sign $0.514 - 0.857i$
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  + (0.167 − 0.290i)2-s + (1.19 + 2.06i)3-s + (0.943 + 1.63i)4-s + (0.5 + 0.866i)5-s + 0.800·6-s + (−2.21 − 3.83i)7-s + 1.30·8-s + (−1.35 + 2.34i)9-s + 0.334·10-s − 2.13·11-s + (−2.25 + 3.90i)12-s + (−0.0906 − 0.156i)13-s − 1.48·14-s + (−1.19 + 2.06i)15-s + (−1.66 + 2.89i)16-s + (2.46 − 4.26i)17-s + ⋯
L(s)  = 1  + (0.118 − 0.205i)2-s + (0.689 + 1.19i)3-s + (0.471 + 0.817i)4-s + (0.223 + 0.387i)5-s + 0.326·6-s + (−0.837 − 1.45i)7-s + 0.460·8-s + (−0.451 + 0.781i)9-s + 0.105·10-s − 0.644·11-s + (−0.650 + 1.12i)12-s + (−0.0251 − 0.0435i)13-s − 0.396·14-s + (−0.308 + 0.534i)15-s + (−0.417 + 0.723i)16-s + (0.597 − 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.514 - 0.857i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ 0.514 - 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37520 + 0.778111i\)
\(L(\frac12)\) \(\approx\) \(1.37520 + 0.778111i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-2.13 - 5.69i)T \)
good2 \( 1 + (-0.167 + 0.290i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.19 - 2.06i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (2.21 + 3.83i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 2.13T + 11T^{2} \)
13 \( 1 + (0.0906 + 0.156i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.46 + 4.26i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.626 + 1.08i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.29T + 23T^{2} \)
29 \( 1 + 1.75T + 29T^{2} \)
31 \( 1 + 3.74T + 31T^{2} \)
41 \( 1 + (1.47 + 2.55i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 12.0T + 43T^{2} \)
47 \( 1 - 0.334T + 47T^{2} \)
53 \( 1 + (-6.01 + 10.4i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.79 - 4.84i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.78 - 10.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.32 + 10.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.95 + 5.12i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 + (2.34 + 4.05i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.60 + 6.25i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.59 - 11.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.49T + 97T^{2} \)
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

−13.10243319899483560999825752287, −11.58434858146220658964378433684, −10.51699855761389688000452139494, −10.05447516157587445342264168081, −8.930561138213040103529654655689, −7.57345671137247725273669188956, −6.82593141908731798695670648470, −4.80607502602475473077175961341, −3.56264921086112335179284918566, −2.96482769735574168976703878192, 1.73797614666267442176078988048, 2.85705018877607305505293746622, 5.36333906999103206866050927084, 6.14246671959607258732147540614, 7.18061785355071770874355865942, 8.362728294966677285727157263416, 9.255703243976015241446254483435, 10.37045897911461554222063153147, 11.75770323962846774529278403000, 12.84636184725181161111338545421

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