↑

Properties

Label 2-1520-380.227-c0-0-3
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  + (0.5 + 0.866i)5-s + (0.366 − 0.366i)7-s + i·9-s − i·11-s + (−0.366 − 0.366i)17-s + 19-s + (1 + i)23-s + (−0.499 + 0.866i)25-s + (0.5 + 0.133i)35-s + (−1.36 − 1.36i)43-s + (−0.866 + 0.5i)45-s + (−1.36 + 1.36i)47-s + 0.732i·49-s + (0.866 − 0.5i)55-s + 1.73·61-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)5-s + (0.366 − 0.366i)7-s + i·9-s − i·11-s + (−0.366 − 0.366i)17-s + 19-s + (1 + i)23-s + (−0.499 + 0.866i)25-s + (0.5 + 0.133i)35-s + (−1.36 − 1.36i)43-s + (−0.866 + 0.5i)45-s + (−1.36 + 1.36i)47-s + 0.732i·49-s + (0.866 − 0.5i)55-s + 1.73·61-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

\(L(\frac{1}{2})\) \(\approx\) \(1.264335351\)
\(L(\frac12)\) \(\approx\) \(1.264335351\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 - T \)
good3 \( 1 - iT^{2} \)
7 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
11 \( 1 + iT - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
23 \( 1 + (-1 - i)T + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
47 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.73T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
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

−9.817608860228561257113700539817, −8.986317823236952822168167061825, −7.989663747508972913229869914833, −7.35851844003853187927265267282, −6.56866346687437759431479890466, −5.53324367477362190982286766267, −4.94732100639917114417961783270, −3.55439615593426913374822201543, −2.77564194266699132194830809074, −1.54691880764820785908254786574, 1.21996930525038273687191638737, 2.35373608845964815625769229982, 3.66500139132664363698558235119, 4.76898383758079044363016755029, 5.28330708656914217079600670813, 6.40629703885201164653779293409, 7.02823598250646004262887351445, 8.282307475979374999281578308918, 8.736748236681472471064747781239, 9.721891443196209021047848430549

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