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Properties

Label 2-3520-440.109-c0-0-11
Degree $2$
Conductor $3520$
Sign $0.707 + 0.707i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $0.707 + 0.707i$
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: \((2,\ 3520,\ (\ :0),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.175296471\)
\(L(\frac12)\) \(\approx\) \(2.175296471\)
\(L(1)\) 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

−8.767946256950164302076180222267, −8.048558279582825154427591682750, −7.56361511265236621340165895609, −6.64299373312283282486141171102, −5.50962454673223641017402143337, −4.64766883754280534621890850430, −3.71602318438917316121262043840, −3.29877118772993937867385549272, −2.22573415631213728104936592697, −1.15136819122058860525173220414, 1.67156152505634684389938739078, 2.69809653214037509243533049961, 3.03992528068649851523868186202, 4.20087684591919318213645080005, 4.52999833712647024950242843477, 6.09861573458229795029337834912, 7.00760643346741468056094504877, 7.39430565432611227099064298834, 8.170110581979348267779280464732, 8.677112701322599045772935620080

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