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Properties

Label 2-1520-380.303-c0-0-5
Degree $2$
Conductor $1520$
Sign $-0.0299 + 0.999i$
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

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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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0299 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0299 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.0299 + 0.999i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :0),\ -0.0299 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.027952707\)
\(L(\frac12)\) \(\approx\) \(1.027952707\)
\(L(1)\) 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 \( 1 + iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.346615549358828817863754209281, −8.778431526336119632838472261702, −8.053536682626320029754828725457, −6.90444707531594704087674833851, −6.05854188676494843219046877587, −5.56986907657189080524550175164, −4.21440093897340253022766992983, −3.63995718415112392048800341318, −2.21508421632310889293862330603, −0.811533867645804144542630610230, 2.15321984517466962522476905279, 2.50676064991783948939205300226, 3.98211152832486381040168214490, 4.88229295106553967430353283597, 5.90928448791373290753344719820, 6.61549018060701743985103103810, 7.38706915222616316747542392671, 8.211697489727363970235403676280, 9.195287843303957661920712544942, 9.952342543339340078308453135336

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