↑

Properties

Label 2-950-95.53-c1-0-23
Degree $2$
Conductor $950$
Sign $-0.999 + 0.0333i$
Analytic cond. $7.58578$
Root an. cond. $2.75423$
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  + (−0.422 − 0.906i)2-s + (−0.284 − 0.199i)3-s + (−0.642 + 0.766i)4-s + (−0.0603 + 0.342i)6-s + (−0.127 − 0.474i)7-s + (0.965 + 0.258i)8-s + (−0.984 − 2.70i)9-s + (−2.42 − 4.19i)11-s + (0.335 − 0.0898i)12-s + (2.60 + 3.72i)13-s + (−0.376 + 0.316i)14-s + (−0.173 − 0.984i)16-s + (5.97 − 2.78i)17-s + (−2.03 + 2.03i)18-s + (−3.35 + 2.78i)19-s + ⋯
L(s)  = 1  + (−0.298 − 0.640i)2-s + (−0.164 − 0.115i)3-s + (−0.321 + 0.383i)4-s + (−0.0246 + 0.139i)6-s + (−0.0481 − 0.179i)7-s + (0.341 + 0.0915i)8-s + (−0.328 − 0.901i)9-s + (−0.730 − 1.26i)11-s + (0.0968 − 0.0259i)12-s + (0.723 + 1.03i)13-s + (−0.100 + 0.0844i)14-s + (−0.0434 − 0.246i)16-s + (1.44 − 0.676i)17-s + (−0.479 + 0.479i)18-s + (−0.768 + 0.639i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.999 + 0.0333i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.999 + 0.0333i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0107555 - 0.644391i\)
\(L(\frac12)\) \(\approx\) \(0.0107555 - 0.644391i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.422 + 0.906i)T \)
5 \( 1 \)
19 \( 1 + (3.35 - 2.78i)T \)
good3 \( 1 + (0.284 + 0.199i)T + (1.02 + 2.81i)T^{2} \)
7 \( 1 + (0.127 + 0.474i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (2.42 + 4.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.60 - 3.72i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (-5.97 + 2.78i)T + (10.9 - 13.0i)T^{2} \)
23 \( 1 + (-0.0964 - 1.10i)T + (-22.6 + 3.99i)T^{2} \)
29 \( 1 + (5.74 - 2.08i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (7.25 + 4.18i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.26 + 7.26i)T + 37iT^{2} \)
41 \( 1 + (-5.95 + 1.05i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (8.97 + 0.785i)T + (42.3 + 7.46i)T^{2} \)
47 \( 1 + (-2.11 + 4.53i)T + (-30.2 - 36.0i)T^{2} \)
53 \( 1 + (7.76 - 0.679i)T + (52.1 - 9.20i)T^{2} \)
59 \( 1 + (5.32 + 1.93i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (10.6 + 8.94i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-0.563 - 0.262i)T + (43.0 + 51.3i)T^{2} \)
71 \( 1 + (2.13 + 2.54i)T + (-12.3 + 69.9i)T^{2} \)
73 \( 1 + (0.412 - 0.588i)T + (-24.9 - 68.5i)T^{2} \)
79 \( 1 + (-1.35 - 7.66i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-8.99 + 2.41i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-0.290 + 1.64i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-6.58 - 14.1i)T + (-62.3 + 74.3i)T^{2} \)
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

−9.513114123328098503188475103642, −8.994841216937186570407847105787, −8.104463303788510163550551037608, −7.20720555916060269196162211812, −6.05515637363592686288894338922, −5.38313086686278430989656143216, −3.77734517082269132861310662612, −3.33512645220630444225689807184, −1.76508204730704698042932085336, −0.33627933684699021407670159108, 1.73974982586183967246769156735, 3.15068956499987279953616504726, 4.57444248905836300970928029253, 5.38196449441450905405230544751, 6.03984163530059263800971469511, 7.30410315610592707019611761049, 7.87312181975743876231156729684, 8.598839470067503695691707464605, 9.629458424894492037585719488434, 10.56634549772329635908387477603

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