↑

Properties

Label 2-1520-380.227-c0-0-5
Degree $2$
Conductor $1520$
Sign $-0.850 + 0.525i$
Analytic cond. $0.758578$
Root an. cond. $0.870964$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + (−1 + i)7-s + i·9-s − 2i·11-s + (−1 − i)17-s − 19-s + (−1 − i)23-s + 25-s + (1 − i)35-s + (−1 − i)43-s − i·45-s + (−1 + i)47-s − i·49-s + 2i·55-s + (−1 − i)63-s + ⋯
L(s)  = 1  − 5-s + (−1 + i)7-s + i·9-s − 2i·11-s + (−1 − i)17-s − 19-s + (−1 − i)23-s + 25-s + (1 − i)35-s + (−1 − i)43-s − i·45-s + (−1 + i)47-s − i·49-s + 2i·55-s + (−1 − i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.850 + 0.525i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :0),\ -0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1574417046\)
\(L(\frac12)\) \(\approx\) \(0.1574417046\)
\(L(1)\) 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 \( 1 + T \)
19 \( 1 + T \)
good3 \( 1 - iT^{2} \)
7 \( 1 + (1 - i)T - iT^{2} \)
11 \( 1 + 2iT - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (1 + i)T + iT^{2} \)
23 \( 1 + (1 + i)T + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (1 + i)T + iT^{2} \)
47 \( 1 + (1 - i)T - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1 - i)T - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-1 - i)T + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - iT^{2} \)
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

−9.037796825837554679896409135332, −8.535535961064254049084133086878, −7.994424197497949672249378346468, −6.76867447766881209153009693405, −6.15359261372446369211809945961, −5.21446220821237168750153853322, −4.19756510683087368494574181698, −3.14575578098784438975102445003, −2.42645605060394033251547545091, −0.11816484723804546267953172115, 1.82132589030254744590015123213, 3.39484842637961466962007140409, 4.06364373982890897506041390110, 4.62404836060009391070166326612, 6.28939079226152775641883089934, 6.79696464998326685828438732061, 7.44153226272227539232397049818, 8.332466617972847881776392176232, 9.325481639633537466705296745468, 9.994044608045269167934295241804

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