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Properties

Label 2-3520-440.189-c0-0-3
Degree $2$
Conductor $3520$
Sign $-0.900 + 0.434i$
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  + (0.309 + 0.951i)5-s + (−0.951 − 0.690i)7-s + (−0.309 + 0.951i)9-s + (0.587 − 0.809i)11-s + (−1.80 − 0.587i)13-s + (−1.53 + 1.11i)19-s − 1.61i·23-s + (−0.809 + 0.587i)25-s + (0.363 − 1.11i)35-s + (−0.5 − 0.363i)37-s + (−0.690 − 0.951i)41-s − 0.999·45-s + (−0.363 − 0.5i)47-s + (0.118 + 0.363i)49-s + (−0.5 + 1.53i)53-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)5-s + (−0.951 − 0.690i)7-s + (−0.309 + 0.951i)9-s + (0.587 − 0.809i)11-s + (−1.80 − 0.587i)13-s + (−1.53 + 1.11i)19-s − 1.61i·23-s + (−0.809 + 0.587i)25-s + (0.363 − 1.11i)35-s + (−0.5 − 0.363i)37-s + (−0.690 − 0.951i)41-s − 0.999·45-s + (−0.363 − 0.5i)47-s + (0.118 + 0.363i)49-s + (−0.5 + 1.53i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $-0.900 + 0.434i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (3489, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3520,\ (\ :0),\ -0.900 + 0.434i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08634024136\)
\(L(\frac12)\) \(\approx\) \(0.08634024136\)
\(L(1)\) 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 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-0.587 + 0.809i)T \)
good3 \( 1 + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + 1.61iT - T^{2} \)
29 \( 1 + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (0.690 + 0.951i)T + (-0.309 + 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 - 1.61T + T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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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

−8.344085616291124591863868401332, −7.62114667814236400546327815148, −6.87965719442295528130150732098, −6.31832571328535408389832896029, −5.57015908407817931651899962362, −4.53409651158063403593163881504, −3.63703840961434413446293865198, −2.77824653911763807307119551969, −2.08191842264885314374128726280, −0.04486708043886707124815792540, 1.72166063525341123343706409228, 2.58537468832784104712722878660, 3.64740821405852015648771336528, 4.65423658848805448109996758848, 5.14970145402436156374540998780, 6.28797110387590842193908579796, 6.63281783702625441002335332493, 7.51962951055761828212354483895, 8.614861917888501592962534066601, 9.205147186619553287875954613262

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