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Properties

Label 2-3584-448.237-c0-0-1
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 + (−1.63 − 1.08i)11-s + (0.707 + 0.292i)23-s + (−0.923 + 0.382i)25-s + (−0.324 + 0.216i)29-s + (−1.63 + 0.324i)37-s + (−0.923 + 1.38i)43-s + (−0.707 + 0.707i)49-s + (−0.923 − 0.617i)53-s + 63-s + (−0.617 − 0.923i)67-s + (−0.541 − 1.30i)71-s + (−0.382 + 1.92i)77-s + (−1.30 + 1.30i)79-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)7-s + (−0.382 + 0.923i)9-s + (−1.63 − 1.08i)11-s + (0.707 + 0.292i)23-s + (−0.923 + 0.382i)25-s + (−0.324 + 0.216i)29-s + (−1.63 + 0.324i)37-s + (−0.923 + 1.38i)43-s + (−0.707 + 0.707i)49-s + (−0.923 − 0.617i)53-s + 63-s + (−0.617 − 0.923i)67-s + (−0.541 − 1.30i)71-s + (−0.382 + 1.92i)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: $\chi_{3584} (2337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ -0.995 - 0.0980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07622308107\)
\(L(\frac12)\) \(\approx\) \(0.07622308107\)
\(L(1)\) 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 \( 1 + (1.63 + 1.08i)T + (0.382 + 0.923i)T^{2} \)
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 \( 1 + (0.324 - 0.216i)T + (0.382 - 0.923i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (1.63 - 0.324i)T + (0.923 - 0.382i)T^{2} \)
41 \( 1 + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + (0.923 - 1.38i)T + (-0.382 - 0.923i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (0.923 + 0.617i)T + (0.382 + 0.923i)T^{2} \)
59 \( 1 + (0.923 - 0.382i)T^{2} \)
61 \( 1 + (0.382 - 0.923i)T^{2} \)
67 \( 1 + (0.617 + 0.923i)T + (-0.382 + 0.923i)T^{2} \)
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

−8.135665737138150986015284509213, −7.76986214010241196719825771542, −6.99633614331423540647865670871, −6.01897764665087254095402628712, −5.29696469899440133083283734571, −4.68923058375528641411177510177, −3.41732769081169518564636350731, −2.95107037858127961539664201830, −1.68889835546375257805968472714, −0.04012982673344177023518815896, 1.94318244852493235812751151405, 2.73017174661721366016961377905, 3.54053310899835714143094394250, 4.68553276422497438522889941623, 5.42978778639528002419608337696, 6.02107684136461147118248047226, 6.96443413800452617419287539334, 7.56621794535098661595135732617, 8.571668415883660601544640901366, 8.951090653606485188590253684944

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