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Properties

Label 2-3520-3520.989-c0-0-0
Degree $2$
Conductor $3520$
Sign $-0.471 - 0.881i$
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  + (−0.290 − 0.956i)2-s + (−0.831 + 0.555i)4-s + (−0.555 + 0.831i)5-s + (−1.42 + 0.591i)7-s + (0.773 + 0.634i)8-s + (0.923 + 0.382i)9-s + (0.956 + 0.290i)10-s + (0.195 + 0.980i)11-s + (0.979 + 1.46i)13-s + (0.980 + 1.19i)14-s + (0.382 − 0.923i)16-s + (−1.40 − 1.40i)17-s + (0.0980 − 0.995i)18-s − i·20-s + (0.881 − 0.471i)22-s + ⋯
L(s)  = 1  + (−0.290 − 0.956i)2-s + (−0.831 + 0.555i)4-s + (−0.555 + 0.831i)5-s + (−1.42 + 0.591i)7-s + (0.773 + 0.634i)8-s + (0.923 + 0.382i)9-s + (0.956 + 0.290i)10-s + (0.195 + 0.980i)11-s + (0.979 + 1.46i)13-s + (0.980 + 1.19i)14-s + (0.382 − 0.923i)16-s + (−1.40 − 1.40i)17-s + (0.0980 − 0.995i)18-s − i·20-s + (0.881 − 0.471i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $-0.471 - 0.881i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (989, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3520,\ (\ :0),\ -0.471 - 0.881i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4475153140\)
\(L(\frac12)\) \(\approx\) \(0.4475153140\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.290 + 0.956i)T \)
5 \( 1 + (0.555 - 0.831i)T \)
11 \( 1 + (-0.195 - 0.980i)T \)
good3 \( 1 + (-0.923 - 0.382i)T^{2} \)
7 \( 1 + (1.42 - 0.591i)T + (0.707 - 0.707i)T^{2} \)
13 \( 1 + (-0.979 - 1.46i)T + (-0.382 + 0.923i)T^{2} \)
17 \( 1 + (1.40 + 1.40i)T + iT^{2} \)
19 \( 1 + (-0.382 + 0.923i)T^{2} \)
23 \( 1 + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.923 + 0.382i)T^{2} \)
31 \( 1 - 0.390iT - T^{2} \)
37 \( 1 + (-0.382 - 0.923i)T^{2} \)
41 \( 1 + (0.707 + 0.707i)T^{2} \)
43 \( 1 + (1.87 - 0.373i)T + (0.923 - 0.382i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (0.923 - 0.382i)T^{2} \)
59 \( 1 + (0.617 - 0.923i)T + (-0.382 - 0.923i)T^{2} \)
61 \( 1 + (-0.923 - 0.382i)T^{2} \)
67 \( 1 + (-0.923 - 0.382i)T^{2} \)
71 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (-0.536 - 0.222i)T + (0.707 + 0.707i)T^{2} \)
79 \( 1 + iT^{2} \)
83 \( 1 + (-1.46 + 0.979i)T + (0.382 - 0.923i)T^{2} \)
89 \( 1 + (0.636 + 1.53i)T + (-0.707 + 0.707i)T^{2} \)
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

−9.188678630654029109740807889341, −8.570884124667283688212375657024, −7.32527722506041826423386950206, −6.92571285665667791507532412555, −6.30559642560615473916148553870, −4.76308373327397311993319103088, −4.23807395004156536017558574119, −3.39753952726327174930475495011, −2.56270170102898753313024657323, −1.74461929536266245909799912786, 0.31368474190786707849346824640, 1.32664259410066147926879424732, 3.56482304477746850249974295675, 3.72606619610153783264504012006, 4.71457596425648759183211515795, 5.79840347117578587636852755419, 6.34991215422452935210400252241, 6.89870591855853126760229623174, 7.928133093525749359687527736878, 8.392436138485278284610283021822

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