↑

Properties

Label 2-1205-5.4-c1-0-5
Degree $2$
Conductor $1205$
Sign $0.287 + 0.957i$
Analytic cond. $9.62197$
Root an. cond. $3.10193$
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  + 1.49i·2-s + 3.12i·3-s − 0.226·4-s + (0.643 + 2.14i)5-s − 4.66·6-s − 4.66i·7-s + 2.64i·8-s − 6.77·9-s + (−3.19 + 0.959i)10-s − 5.92·11-s − 0.707i·12-s − 4.29i·13-s + 6.95·14-s + (−6.69 + 2.01i)15-s − 4.40·16-s + 5.72i·17-s + ⋯
L(s)  = 1  + 1.05i·2-s + 1.80i·3-s − 0.113·4-s + (0.287 + 0.957i)5-s − 1.90·6-s − 1.76i·7-s + 0.935i·8-s − 2.25·9-s + (−1.01 + 0.303i)10-s − 1.78·11-s − 0.204i·12-s − 1.19i·13-s + 1.85·14-s + (−1.72 + 0.519i)15-s − 1.10·16-s + 1.38i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $0.287 + 0.957i$
Analytic conductor: \(9.62197\)
Root analytic conductor: \(3.10193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (724, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1205,\ (\ :1/2),\ 0.287 + 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8468533724\)
\(L(\frac12)\) \(\approx\) \(0.8468533724\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.643 - 2.14i)T \)
241 \( 1 - T \)
good2 \( 1 - 1.49iT - 2T^{2} \)
3 \( 1 - 3.12iT - 3T^{2} \)
7 \( 1 + 4.66iT - 7T^{2} \)
11 \( 1 + 5.92T + 11T^{2} \)
13 \( 1 + 4.29iT - 13T^{2} \)
17 \( 1 - 5.72iT - 17T^{2} \)
19 \( 1 - 1.23T + 19T^{2} \)
23 \( 1 + 3.63iT - 23T^{2} \)
29 \( 1 + 1.81T + 29T^{2} \)
31 \( 1 + 1.11T + 31T^{2} \)
37 \( 1 - 7.32iT - 37T^{2} \)
41 \( 1 + 6.45T + 41T^{2} \)
43 \( 1 + 4.57iT - 43T^{2} \)
47 \( 1 - 7.71iT - 47T^{2} \)
53 \( 1 - 10.1iT - 53T^{2} \)
59 \( 1 - 5.66T + 59T^{2} \)
61 \( 1 - 4.76T + 61T^{2} \)
67 \( 1 - 14.0iT - 67T^{2} \)
71 \( 1 + 3.96T + 71T^{2} \)
73 \( 1 - 8.92iT - 73T^{2} \)
79 \( 1 + 1.77T + 79T^{2} \)
83 \( 1 + 9.93iT - 83T^{2} \)
89 \( 1 - 7.03T + 89T^{2} \)
97 \( 1 + 5.03iT - 97T^{2} \)
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

−10.28440567917055024452973684027, −10.08721031000480498547335149214, −8.506860370132281647774398998431, −7.892711331196675785723226264732, −7.17702461207128941741751148307, −6.06680757585855082956387668124, −5.40207473758947179091539145512, −4.54458791156347536095652274335, −3.52560507458362549472542007446, −2.71423930361169256715003286344, 0.32003534494086886500265270771, 1.88967047088151670610930750909, 2.16989503134317547878853507792, 3.08040602986410506042178852127, 5.10497446205861890171523162646, 5.59736361548467690852171037367, 6.64053555665116697477348062712, 7.50434651721452957466994483745, 8.341347985679568086119056450667, 9.123668425748821411178496658685

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