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Properties

Label 2-3520-440.109-c0-0-3
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  + 5-s − 9-s + i·11-s + 2i·23-s + 25-s + 2·37-s − 45-s + 2i·47-s − 49-s + 2·53-s + i·55-s − 2i·59-s + 81-s − 2·89-s − i·99-s + ⋯
L(s)  = 1  + 5-s − 9-s + i·11-s + 2i·23-s + 25-s + 2·37-s − 45-s + 2i·47-s − 49-s + 2·53-s + i·55-s − 2i·59-s + 81-s − 2·89-s − i·99-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

\(L(\frac{1}{2})\) \(\approx\) \(1.388986737\)
\(L(\frac12)\) \(\approx\) \(1.388986737\)
\(L(1)\) 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 \( 1 - T \)
11 \( 1 - iT \)
good3 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - 2iT - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - 2T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 2iT - T^{2} \)
53 \( 1 - 2T + T^{2} \)
59 \( 1 + 2iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 2T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.005559472555653922053082752517, −8.070329094907495443903101727773, −7.40278053775260587784322967184, −6.50280070664950291448969087494, −5.79888878149728613208702056219, −5.23051499730194382891164377186, −4.32199084087432987763749201351, −3.17060429241920250141855999752, −2.37242309767012178786020893882, −1.41086516551421474696503783866, 0.855545287532692104444043906997, 2.36903036838174079300672929582, 2.84066631849870274348092896876, 4.00499650238007141415257985472, 5.03145969038554748342396731421, 5.78975668512418259265945515669, 6.24055965565813193421198569056, 7.02289707875300542361370793008, 8.221822575848665255909552140105, 8.621929441664309984554789228796

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