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Properties

Label 2-3520-440.109-c0-0-2
Degree $2$
Conductor $3520$
Sign $-0.258 - 0.965i$
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  − 1.73·3-s + (0.5 + 0.866i)5-s + 1.99·9-s + i·11-s + (−0.866 − 1.49i)15-s + i·23-s + (−0.499 + 0.866i)25-s − 1.73·27-s + 1.73·31-s − 1.73i·33-s + 37-s + (0.999 + 1.73i)45-s − 2i·47-s − 49-s − 2·53-s + ⋯
L(s)  = 1  − 1.73·3-s + (0.5 + 0.866i)5-s + 1.99·9-s + i·11-s + (−0.866 − 1.49i)15-s + i·23-s + (−0.499 + 0.866i)25-s − 1.73·27-s + 1.73·31-s − 1.73i·33-s + 37-s + (0.999 + 1.73i)45-s − 2i·47-s − 49-s − 2·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $-0.258 - 0.965i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (2529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3520,\ (\ :0),\ -0.258 - 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6737899157\)
\(L(\frac12)\) \(\approx\) \(0.6737899157\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 - iT \)
good3 \( 1 + 1.73T + T^{2} \)
7 \( 1 + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - 1.73T + T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + 2iT - T^{2} \)
53 \( 1 + 2T + T^{2} \)
59 \( 1 - iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - 1.73T + T^{2} \)
71 \( 1 + 1.73T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 - 1.73iT - T^{2} \)
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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

−9.306305668428773581077839508843, −7.921717984292566285865423435897, −7.23513500569771574088722268186, −6.51830990700516464331399850176, −6.14138664663234565839052641579, −5.22641791558527257961000003619, −4.68703947410126489780276326190, −3.64647671146135631165209325903, −2.37800606146661769883698567925, −1.30191608608159025271692911023, 0.56412847099716136396397827414, 1.48176916711645414993239425555, 2.93625073961764281989169315032, 4.43579191736324447770456795145, 4.71376094138110027253982012201, 5.65481930558233683436965414802, 6.19819744353842435041723462073, 6.60202256326532454081877881596, 7.86969838890071936224989417237, 8.457712532930896045881165788260

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