Properties

Label 2-3584-448.13-c0-0-0
Degree $2$
Conductor $3584$
Sign $0.995 + 0.0980i$
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  + (0.382 + 0.923i)7-s + (0.382 − 0.923i)9-s + (0.216 − 0.324i)11-s + (0.707 + 0.292i)23-s + (0.923 − 0.382i)25-s + (−1.08 − 1.63i)29-s + (0.216 + 1.08i)37-s + (0.923 + 0.617i)43-s + (−0.707 + 0.707i)49-s + (0.923 − 1.38i)53-s + 63-s + (−1.38 + 0.923i)67-s + (0.541 + 1.30i)71-s + (0.382 + 0.0761i)77-s + (1.30 − 1.30i)79-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)7-s + (0.382 − 0.923i)9-s + (0.216 − 0.324i)11-s + (0.707 + 0.292i)23-s + (0.923 − 0.382i)25-s + (−1.08 − 1.63i)29-s + (0.216 + 1.08i)37-s + (0.923 + 0.617i)43-s + (−0.707 + 0.707i)49-s + (0.923 − 1.38i)53-s + 63-s + (−1.38 + 0.923i)67-s + (0.541 + 1.30i)71-s + (0.382 + 0.0761i)77-s + (1.30 − 1.30i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $0.995 + 0.0980i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ 0.995 + 0.0980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.428071335\)
\(L(\frac12)\) \(\approx\) \(1.428071335\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.382 - 0.923i)T \)
good3 \( 1 + (-0.382 + 0.923i)T^{2} \)
5 \( 1 + (-0.923 + 0.382i)T^{2} \)
11 \( 1 + (-0.216 + 0.324i)T + (-0.382 - 0.923i)T^{2} \)
13 \( 1 + (-0.923 - 0.382i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (0.923 + 0.382i)T^{2} \)
23 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (1.08 + 1.63i)T + (-0.382 + 0.923i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.216 - 1.08i)T + (-0.923 + 0.382i)T^{2} \)
41 \( 1 + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + (-0.923 - 0.617i)T + (0.382 + 0.923i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-0.923 + 1.38i)T + (-0.382 - 0.923i)T^{2} \)
59 \( 1 + (-0.923 + 0.382i)T^{2} \)
61 \( 1 + (-0.382 + 0.923i)T^{2} \)
67 \( 1 + (1.38 - 0.923i)T + (0.382 - 0.923i)T^{2} \)
71 \( 1 + (-0.541 - 1.30i)T + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (0.707 + 0.707i)T^{2} \)
79 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
83 \( 1 + (0.923 + 0.382i)T^{2} \)
89 \( 1 + (-0.707 + 0.707i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.792550516232014720379164751042, −8.079082675234512949329958081312, −7.23001982769493958656622920240, −6.39076554111867216645333168986, −5.83307462936359152132081254850, −4.94643208129582265008506812326, −4.10059929504666672595762881661, −3.18441893669126692677826933961, −2.27122566921199880228700095367, −1.05763875084802447877328725654, 1.17908716964799183519340592230, 2.14015438198263714535510167101, 3.32162454629884592162181488922, 4.22661404294962526313510960421, 4.89276953979563794381355748501, 5.59055870534455300870535396919, 6.79877557685503611690923464811, 7.30985523891772222484157705980, 7.79162247209442995235692434466, 8.836129296315143284447648331670

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