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Properties

Label 2-3520-44.43-c1-0-81
Degree $2$
Conductor $3520$
Sign $-0.904 + 0.426i$
Analytic cond. $28.1073$
Root an. cond. $5.30163$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·3-s + 5-s − 4·7-s + 0.999·9-s + (3 − 1.41i)11-s − 4.24i·13-s − 1.41i·15-s − 1.41i·17-s − 2·19-s + 5.65i·21-s + 7.07i·23-s + 25-s − 5.65i·27-s − 2.82i·29-s − 8.48i·31-s + ⋯
L(s)  = 1  − 0.816i·3-s + 0.447·5-s − 1.51·7-s + 0.333·9-s + (0.904 − 0.426i)11-s − 1.17i·13-s − 0.365i·15-s − 0.342i·17-s − 0.458·19-s + 1.23i·21-s + 1.47i·23-s + 0.200·25-s − 1.08i·27-s − 0.525i·29-s − 1.52i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $-0.904 + 0.426i$
Analytic conductor: \(28.1073\)
Root analytic conductor: \(5.30163\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (2111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3520,\ (\ :1/2),\ -0.904 + 0.426i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.295660543\)
\(L(\frac12)\) \(\approx\) \(1.295660543\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3 + 1.41i)T \)
good3 \( 1 + 1.41iT - 3T^{2} \)
7 \( 1 + 4T + 7T^{2} \)
13 \( 1 + 4.24iT - 13T^{2} \)
17 \( 1 + 1.41iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 7.07iT - 23T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 + 8.48iT - 31T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 - 2.82iT - 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 - 7.07iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 2.82iT - 59T^{2} \)
61 \( 1 + 8.48iT - 61T^{2} \)
67 \( 1 - 4.24iT - 67T^{2} \)
71 \( 1 + 5.65iT - 71T^{2} \)
73 \( 1 + 4.24iT - 73T^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−8.003221123283887914541766775712, −7.56651941089178150800132656085, −6.49617894075015820168505406100, −6.29887937457082838703022484141, −5.54703408153792384400815134388, −4.26478816004000506844697515950, −3.37766797588235592046942317033, −2.62774291852375960533648519291, −1.43957727227369303982034111407, −0.39637858679969621621876243927, 1.41208636245298268140110172989, 2.57278805372048431934767436617, 3.60117952680927010723954107335, 4.20802357661074006711521015741, 4.91273086489893400691573447556, 6.07370892866409056681443658935, 6.76330715335996580324673466526, 6.94448579817377048715639098434, 8.485925820643422940386286056022, 9.128663594818806879450882406006

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