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Properties

Label 2-3520-44.43-c1-0-32
Degree $2$
Conductor $3520$
Sign $-0.982 - 0.185i$
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  + 3.25i·3-s + 5-s + 3.25·7-s − 7.62·9-s + (3.25 + 0.613i)11-s + 2i·13-s + 3.25i·15-s + 4.62i·17-s − 4.48·19-s + 10.6i·21-s + 1.22i·23-s + 25-s − 15.0i·27-s − 2.62i·29-s − 3.25i·31-s + ⋯
L(s)  = 1  + 1.88i·3-s + 0.447·5-s + 1.23·7-s − 2.54·9-s + (0.982 + 0.185i)11-s + 0.554i·13-s + 0.841i·15-s + 1.12i·17-s − 1.02·19-s + 2.31i·21-s + 0.255i·23-s + 0.200·25-s − 2.90i·27-s − 0.487i·29-s − 0.585i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.185i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $-0.982 - 0.185i$
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: \((2,\ 3520,\ (\ :1/2),\ -0.982 - 0.185i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.108483015\)
\(L(\frac12)\) \(\approx\) \(2.108483015\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.25 - 0.613i)T \)
good3 \( 1 - 3.25iT - 3T^{2} \)
7 \( 1 - 3.25T + 7T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 4.62iT - 17T^{2} \)
19 \( 1 + 4.48T + 19T^{2} \)
23 \( 1 - 1.22iT - 23T^{2} \)
29 \( 1 + 2.62iT - 29T^{2} \)
31 \( 1 + 3.25iT - 31T^{2} \)
37 \( 1 - 4.62T + 37T^{2} \)
41 \( 1 - 8iT - 41T^{2} \)
43 \( 1 - 1.22T + 43T^{2} \)
47 \( 1 - 1.22iT - 47T^{2} \)
53 \( 1 + 0.623T + 53T^{2} \)
59 \( 1 - 11.8iT - 59T^{2} \)
61 \( 1 + 1.37iT - 61T^{2} \)
67 \( 1 - 7.74iT - 67T^{2} \)
71 \( 1 + 9.77iT - 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 + 7.74T + 83T^{2} \)
89 \( 1 + 8.62T + 89T^{2} \)
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

−8.857714100916752788915591648096, −8.594096375529683487710050053581, −7.67707546726578791851897196348, −6.31071681668687012159092811839, −5.86269248092670405467781842982, −4.85025998493097913310275690334, −4.30036844931282117962722757084, −3.87477301970754354070284085846, −2.60266788832186207694020306598, −1.55338966736134744426471116452, 0.62595937739503207036531450779, 1.54344576515936925047231696995, 2.19935616458519985295351692598, 3.14867755300315557692585835784, 4.54621404099045154153294872580, 5.45351037188856839109110609572, 6.09042194578395387189076133260, 6.92813203921554330490955440286, 7.30938170944527126629041556437, 8.301559183370091861251838094891

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