↑

Properties

Label 2-1040-20.7-c1-0-5
Degree $2$
Conductor $1040$
Sign $0.147 - 0.989i$
Analytic cond. $8.30444$
Root an. cond. $2.88174$
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  + (−0.246 − 0.246i)3-s + (−1.83 + 1.27i)5-s + (1.16 − 1.16i)7-s − 2.87i·9-s + 4.32i·11-s + (−0.707 + 0.707i)13-s + (0.766 + 0.138i)15-s + (2.78 + 2.78i)17-s − 5.23·19-s − 0.573·21-s + (2.06 + 2.06i)23-s + (1.74 − 4.68i)25-s + (−1.44 + 1.44i)27-s − 1.84i·29-s + 8.65i·31-s + ⋯
L(s)  = 1  + (−0.142 − 0.142i)3-s + (−0.821 + 0.570i)5-s + (0.439 − 0.439i)7-s − 0.959i·9-s + 1.30i·11-s + (−0.196 + 0.196i)13-s + (0.197 + 0.0356i)15-s + (0.674 + 0.674i)17-s − 1.19·19-s − 0.125·21-s + (0.430 + 0.430i)23-s + (0.349 − 0.937i)25-s + (−0.278 + 0.278i)27-s − 0.341i·29-s + 1.55i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.147 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.147 - 0.989i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ 0.147 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.042758955\)
\(L(\frac12)\) \(\approx\) \(1.042758955\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1.83 - 1.27i)T \)
13 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (0.246 + 0.246i)T + 3iT^{2} \)
7 \( 1 + (-1.16 + 1.16i)T - 7iT^{2} \)
11 \( 1 - 4.32iT - 11T^{2} \)
17 \( 1 + (-2.78 - 2.78i)T + 17iT^{2} \)
19 \( 1 + 5.23T + 19T^{2} \)
23 \( 1 + (-2.06 - 2.06i)T + 23iT^{2} \)
29 \( 1 + 1.84iT - 29T^{2} \)
31 \( 1 - 8.65iT - 31T^{2} \)
37 \( 1 + (-3.13 - 3.13i)T + 37iT^{2} \)
41 \( 1 - 12.2T + 41T^{2} \)
43 \( 1 + (1.55 + 1.55i)T + 43iT^{2} \)
47 \( 1 + (8.20 - 8.20i)T - 47iT^{2} \)
53 \( 1 + (8.76 - 8.76i)T - 53iT^{2} \)
59 \( 1 - 8.70T + 59T^{2} \)
61 \( 1 - 2.92T + 61T^{2} \)
67 \( 1 + (10.1 - 10.1i)T - 67iT^{2} \)
71 \( 1 - 8.12iT - 71T^{2} \)
73 \( 1 + (-1.50 + 1.50i)T - 73iT^{2} \)
79 \( 1 + 1.92T + 79T^{2} \)
83 \( 1 + (-8.76 - 8.76i)T + 83iT^{2} \)
89 \( 1 + 8.27iT - 89T^{2} \)
97 \( 1 + (5.37 + 5.37i)T + 97iT^{2} \)
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

−10.19382441120109454838770410504, −9.357405230480021603466254412135, −8.307783849170822920797488015075, −7.50108077309120869978938810507, −6.88244235832713331463489721649, −6.05917208381416112530039403852, −4.61016646957194241290115221572, −4.03924259179426060815613968284, −2.88167348124001788718405791909, −1.36467419378521026049069295663, 0.51751849616454566673316367228, 2.24894146003215256071155803412, 3.49417501536698754714914480134, 4.60484865805069074540344367132, 5.27779787409669555898804719655, 6.18201323221228910522024022300, 7.52589104645206003569965328085, 8.137316891955274932408487606455, 8.702420931047917241506860956322, 9.675035429212053176315230554923

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