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Properties

Label 2-3520-440.29-c0-0-3
Degree $2$
Conductor $3520$
Sign $0.390 + 0.920i$
Analytic cond. $1.75670$
Root an. cond. $1.32540$
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  + (0.809 − 0.587i)5-s + (−0.587 − 1.80i)7-s + (0.809 + 0.587i)9-s + (0.951 − 0.309i)11-s + (0.690 − 0.951i)13-s + (−0.363 + 1.11i)19-s + 0.618i·23-s + (0.309 − 0.951i)25-s + (−1.53 − 1.11i)35-s + (0.5 + 1.53i)37-s + (−1.80 − 0.587i)41-s + 45-s + (−1.53 − 0.5i)47-s + (−2.11 + 1.53i)49-s + (0.5 + 0.363i)53-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)5-s + (−0.587 − 1.80i)7-s + (0.809 + 0.587i)9-s + (0.951 − 0.309i)11-s + (0.690 − 0.951i)13-s + (−0.363 + 1.11i)19-s + 0.618i·23-s + (0.309 − 0.951i)25-s + (−1.53 − 1.11i)35-s + (0.5 + 1.53i)37-s + (−1.80 − 0.587i)41-s + 45-s + (−1.53 − 0.5i)47-s + (−2.11 + 1.53i)49-s + (0.5 + 0.363i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $0.390 + 0.920i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (1569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3520,\ (\ :0),\ 0.390 + 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.585527536\)
\(L(\frac12)\) \(\approx\) \(1.585527536\)
\(L(1)\) 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 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-0.951 + 0.309i)T \)
good3 \( 1 + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.690 + 0.951i)T + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 - 0.618iT - T^{2} \)
29 \( 1 + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.309 - 0.951i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + 0.618T + T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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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

−8.436217666059812598340756193646, −8.005762749782475995414284757461, −6.96106911714570252315393923780, −6.52681173091204338578352765301, −5.64007438610001870104187358505, −4.72923329213562940180422316588, −3.90574726922477277432624165240, −3.34572727922676105772049751080, −1.67353056220311728516967311008, −1.05357708852039984194172878090, 1.63059171100554535819381646973, 2.35925589389760663060398351774, 3.28673965271406362222481851105, 4.23687905022389256593228919142, 5.24822197338010558108078381992, 6.12716062532827025538393458133, 6.62856228867365634955686316706, 6.96128579666728409952220342120, 8.503457768548432462109165383072, 9.003560423026295774945726289651

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