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Properties

Label 2-3584-16.13-c1-0-23
Degree $2$
Conductor $3584$
Sign $0.923 - 0.382i$
Analytic cond. $28.6183$
Root an. cond. $5.34961$
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 − i)3-s + i·7-s − i·9-s + (2 − 2i)11-s + (2 + 2i)13-s − 6·17-s + (3 + 3i)19-s + (1 − i)21-s + 5i·25-s + (−4 + 4i)27-s + (1 + i)29-s + 10·31-s − 4·33-s + (−3 + 3i)37-s − 4i·39-s + ⋯
L(s)  = 1  + (−0.577 − 0.577i)3-s + 0.377i·7-s − 0.333i·9-s + (0.603 − 0.603i)11-s + (0.554 + 0.554i)13-s − 1.45·17-s + (0.688 + 0.688i)19-s + (0.218 − 0.218i)21-s + i·25-s + (−0.769 + 0.769i)27-s + (0.185 + 0.185i)29-s + 1.79·31-s − 0.696·33-s + (−0.493 + 0.493i)37-s − 0.640i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(28.6183\)
Root analytic conductor: \(5.34961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (2689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :1/2),\ 0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.335731576\)
\(L(\frac12)\) \(\approx\) \(1.335731576\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - iT \)
good3 \( 1 + (1 + i)T + 3iT^{2} \)
5 \( 1 - 5iT^{2} \)
11 \( 1 + (-2 + 2i)T - 11iT^{2} \)
13 \( 1 + (-2 - 2i)T + 13iT^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + (-3 - 3i)T + 19iT^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (-1 - i)T + 29iT^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 + 10iT - 41T^{2} \)
43 \( 1 + (6 - 6i)T - 43iT^{2} \)
47 \( 1 - 2T + 47T^{2} \)
53 \( 1 + (9 - 9i)T - 53iT^{2} \)
59 \( 1 + (-1 + i)T - 59iT^{2} \)
61 \( 1 + 61iT^{2} \)
67 \( 1 + (-8 - 8i)T + 67iT^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + (-5 - 5i)T + 83iT^{2} \)
89 \( 1 - 2iT - 89T^{2} \)
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

−8.720896169802069549500431596238, −7.83556395751819059098818767534, −6.83034365213455582497997607639, −6.44542692879828832960847636953, −5.81995922073907489116130674111, −4.90184678807915357556342563513, −3.91991217738304597148873938062, −3.11474292986227229139148086756, −1.82859481496695112388448128112, −0.955211197001957626652341717521, 0.53521951332923409770780734381, 1.92591551945622809245006133727, 2.99714338988134201988406166770, 4.15538514599008159235711105536, 4.62989532416718160714617783930, 5.33149515701599302903805608781, 6.45204689783399799645296987877, 6.74047755614862855159109838884, 7.889857796359656144916483622654, 8.445717368368108801067295166403

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