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Properties

Label 2-3584-3584.1357-c0-0-0
Degree $2$
Conductor $3584$
Sign $0.838 - 0.545i$
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  + (−0.336 + 0.941i)2-s + (−0.773 − 0.634i)4-s + (0.970 − 0.242i)7-s + (0.857 − 0.514i)8-s + (0.989 − 0.146i)9-s + (−1.88 − 0.520i)11-s + (−0.0980 + 0.995i)14-s + (0.195 + 0.980i)16-s + (−0.195 + 0.980i)18-s + (1.12 − 1.59i)22-s + (0.541 + 1.51i)23-s + (−0.0490 − 0.998i)25-s + (−0.903 − 0.427i)28-s + (1.23 + 1.57i)29-s + (−0.989 − 0.146i)32-s + ⋯
L(s)  = 1  + (−0.336 + 0.941i)2-s + (−0.773 − 0.634i)4-s + (0.970 − 0.242i)7-s + (0.857 − 0.514i)8-s + (0.989 − 0.146i)9-s + (−1.88 − 0.520i)11-s + (−0.0980 + 0.995i)14-s + (0.195 + 0.980i)16-s + (−0.195 + 0.980i)18-s + (1.12 − 1.59i)22-s + (0.541 + 1.51i)23-s + (−0.0490 − 0.998i)25-s + (−0.903 − 0.427i)28-s + (1.23 + 1.57i)29-s + (−0.989 − 0.146i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $0.838 - 0.545i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (1357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ 0.838 - 0.545i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.100323124\)
\(L(\frac12)\) \(\approx\) \(1.100323124\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.336 - 0.941i)T \)
7 \( 1 + (-0.970 + 0.242i)T \)
good3 \( 1 + (-0.989 + 0.146i)T^{2} \)
5 \( 1 + (0.0490 + 0.998i)T^{2} \)
11 \( 1 + (1.88 + 0.520i)T + (0.857 + 0.514i)T^{2} \)
13 \( 1 + (0.740 - 0.671i)T^{2} \)
17 \( 1 + (-0.195 + 0.980i)T^{2} \)
19 \( 1 + (-0.903 - 0.427i)T^{2} \)
23 \( 1 + (-0.541 - 1.51i)T + (-0.773 + 0.634i)T^{2} \)
29 \( 1 + (-1.23 - 1.57i)T + (-0.242 + 0.970i)T^{2} \)
31 \( 1 + (-0.923 + 0.382i)T^{2} \)
37 \( 1 + (-0.329 + 1.89i)T + (-0.941 - 0.336i)T^{2} \)
41 \( 1 + (0.995 + 0.0980i)T^{2} \)
43 \( 1 + (0.159 + 0.185i)T + (-0.146 + 0.989i)T^{2} \)
47 \( 1 + (0.555 + 0.831i)T^{2} \)
53 \( 1 + (-0.553 + 0.708i)T + (-0.242 - 0.970i)T^{2} \)
59 \( 1 + (-0.740 - 0.671i)T^{2} \)
61 \( 1 + (0.803 + 0.595i)T^{2} \)
67 \( 1 + (-0.680 - 1.35i)T + (-0.595 + 0.803i)T^{2} \)
71 \( 1 + (-1.53 + 1.14i)T + (0.290 - 0.956i)T^{2} \)
73 \( 1 + (-0.881 - 0.471i)T^{2} \)
79 \( 1 + (-0.389 + 1.28i)T + (-0.831 - 0.555i)T^{2} \)
83 \( 1 + (0.941 - 0.336i)T^{2} \)
89 \( 1 + (0.773 + 0.634i)T^{2} \)
97 \( 1 + (-0.382 - 0.923i)T^{2} \)
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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

−8.556533263273384406434527627390, −7.963664691696670425010983669604, −7.41134625550599635110781651864, −6.83215254878579236577243507759, −5.67979678730754161695109519733, −5.16160503610436050901424894390, −4.53228119865718867350705377987, −3.50464343057509464837020426089, −2.11620499927295977453012274198, −0.921653163903062301368605378016, 1.10214802386531340566998431893, 2.26455715299091701985802083746, 2.73742990821756138280575584958, 4.09967172826175499826780094508, 4.85534681555484690572804120325, 5.14739047354982540285338122926, 6.58132936337642159502063491910, 7.58944743020051623747569677528, 8.021255027691681504279346039829, 8.559634283356748642255994064375

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