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Properties

Label 2-3584-3584.2421-c0-0-0
Degree $2$
Conductor $3584$
Sign $0.278 - 0.960i$
Analytic cond. $1.78864$
Root an. cond. $1.33740$
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  + (0.803 + 0.595i)2-s + (0.290 + 0.956i)4-s + (−0.427 + 0.903i)7-s + (−0.336 + 0.941i)8-s + (0.998 − 0.0490i)9-s + (1.14 − 1.63i)11-s + (−0.881 + 0.471i)14-s + (−0.831 + 0.555i)16-s + (0.831 + 0.555i)18-s + (1.89 − 0.625i)22-s + (1.07 − 0.800i)23-s + (0.857 − 0.514i)25-s + (−0.989 − 0.146i)28-s + (0.215 + 0.961i)29-s + (−0.998 − 0.0490i)32-s + ⋯
L(s)  = 1  + (0.803 + 0.595i)2-s + (0.290 + 0.956i)4-s + (−0.427 + 0.903i)7-s + (−0.336 + 0.941i)8-s + (0.998 − 0.0490i)9-s + (1.14 − 1.63i)11-s + (−0.881 + 0.471i)14-s + (−0.831 + 0.555i)16-s + (0.831 + 0.555i)18-s + (1.89 − 0.625i)22-s + (1.07 − 0.800i)23-s + (0.857 − 0.514i)25-s + (−0.989 − 0.146i)28-s + (0.215 + 0.961i)29-s + (−0.998 − 0.0490i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $0.278 - 0.960i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (2421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ 0.278 - 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.227983895\)
\(L(\frac12)\) \(\approx\) \(2.227983895\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.803 - 0.595i)T \)
7 \( 1 + (0.427 - 0.903i)T \)
good3 \( 1 + (-0.998 + 0.0490i)T^{2} \)
5 \( 1 + (-0.857 + 0.514i)T^{2} \)
11 \( 1 + (-1.14 + 1.63i)T + (-0.336 - 0.941i)T^{2} \)
13 \( 1 + (0.970 - 0.242i)T^{2} \)
17 \( 1 + (0.831 + 0.555i)T^{2} \)
19 \( 1 + (-0.989 - 0.146i)T^{2} \)
23 \( 1 + (-1.07 + 0.800i)T + (0.290 - 0.956i)T^{2} \)
29 \( 1 + (-0.215 - 0.961i)T + (-0.903 + 0.427i)T^{2} \)
31 \( 1 + (0.382 - 0.923i)T^{2} \)
37 \( 1 + (1.02 - 0.517i)T + (0.595 - 0.803i)T^{2} \)
41 \( 1 + (-0.471 - 0.881i)T^{2} \)
43 \( 1 + (1.22 - 1.16i)T + (0.0490 - 0.998i)T^{2} \)
47 \( 1 + (-0.195 - 0.980i)T^{2} \)
53 \( 1 + (0.408 - 1.82i)T + (-0.903 - 0.427i)T^{2} \)
59 \( 1 + (-0.970 - 0.242i)T^{2} \)
61 \( 1 + (-0.671 + 0.740i)T^{2} \)
67 \( 1 + (1.28 + 0.496i)T + (0.740 + 0.671i)T^{2} \)
71 \( 1 + (1.33 + 1.47i)T + (-0.0980 + 0.995i)T^{2} \)
73 \( 1 + (0.634 - 0.773i)T^{2} \)
79 \( 1 + (-0.0476 + 0.483i)T + (-0.980 - 0.195i)T^{2} \)
83 \( 1 + (-0.595 - 0.803i)T^{2} \)
89 \( 1 + (-0.290 - 0.956i)T^{2} \)
97 \( 1 + (-0.923 - 0.382i)T^{2} \)
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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

−8.901087374153088399082248383660, −8.173548945419554493879052164138, −7.11880900342986539301653634654, −6.45745986624168321786154163851, −6.11949981125297456672220272790, −5.07462672003298444033386507857, −4.44201223419739731111277223474, −3.31531119684530620926513498704, −2.97104328664913127094931506345, −1.44073596919331247313632739280, 1.23311566180158444325185888537, 1.92909758223065610027628329124, 3.28348282419300503762232626284, 3.98564542534075536011555998319, 4.56415561045886630590913795275, 5.28621349580090201066369047184, 6.53753190257282131746720299186, 7.03552095541102312039029661720, 7.33917772909827459504778758014, 8.857458843983142626012926055082

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