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Properties

Label 2-1520-380.227-c0-0-4
Degree $2$
Conductor $1520$
Sign $0.880 + 0.473i$
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 + (1.36 − 1.36i)7-s + i·9-s + i·11-s + (1.36 + 1.36i)17-s − 19-s + (−1 − i)23-s + (−0.499 − 0.866i)25-s + (−0.499 − 1.86i)35-s + (−0.366 − 0.366i)43-s + (0.866 + 0.5i)45-s + (−0.366 + 0.366i)47-s − 2.73i·49-s + (0.866 + 0.5i)55-s − 1.73·61-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)5-s + (1.36 − 1.36i)7-s + i·9-s + i·11-s + (1.36 + 1.36i)17-s − 19-s + (−1 − i)23-s + (−0.499 − 0.866i)25-s + (−0.499 − 1.86i)35-s + (−0.366 − 0.366i)43-s + (0.866 + 0.5i)45-s + (−0.366 + 0.366i)47-s − 2.73i·49-s + (0.866 + 0.5i)55-s − 1.73·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.880 + 0.473i$
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: \((2,\ 1520,\ (\ :0),\ 0.880 + 0.473i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.377593337\)
\(L(\frac12)\) \(\approx\) \(1.377593337\)
\(L(1)\) 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 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
11 \( 1 - iT - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
23 \( 1 + (1 + i)T + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
47 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.73T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
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

−9.835757955599820517844183907969, −8.490559635239891675092046055370, −8.056055536386904650610756955396, −7.47243271955041965929918022598, −6.29780670700474426352865053909, −5.26012630926880406801584863370, −4.53842893572632213533274855621, −4.03549405888289826068462637111, −2.05382391842168764844366881365, −1.44272402410046426389380810607, 1.59971254040730101443022412759, 2.71431181034684292577889244386, 3.51542651371191792976577038865, 4.93740020012881673606928530317, 5.79657530338218944313481158632, 6.18585583063611082981765665398, 7.42105900736874725345110192512, 8.144840501983645169099232745441, 8.996889447906535091184268758758, 9.590893855706105472963146962720

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