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Properties

Label 2-3584-448.181-c0-0-1
Degree $2$
Conductor $3584$
Sign $0.773 + 0.634i$
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  + (0.923 + 0.382i)7-s + (0.923 − 0.382i)9-s + (0.324 − 1.63i)11-s + (−0.707 − 1.70i)23-s + (−0.382 + 0.923i)25-s + (−0.216 − 1.08i)29-s + (0.324 − 0.216i)37-s + (−0.382 − 0.0761i)43-s + (0.707 + 0.707i)49-s + (−0.382 + 1.92i)53-s + 63-s + (−1.92 + 0.382i)67-s + (−1.30 − 0.541i)71-s + (0.923 − 1.38i)77-s + (0.541 + 0.541i)79-s + ⋯
L(s)  = 1  + (0.923 + 0.382i)7-s + (0.923 − 0.382i)9-s + (0.324 − 1.63i)11-s + (−0.707 − 1.70i)23-s + (−0.382 + 0.923i)25-s + (−0.216 − 1.08i)29-s + (0.324 − 0.216i)37-s + (−0.382 − 0.0761i)43-s + (0.707 + 0.707i)49-s + (−0.382 + 1.92i)53-s + 63-s + (−1.92 + 0.382i)67-s + (−1.30 − 0.541i)71-s + (0.923 − 1.38i)77-s + (0.541 + 0.541i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $0.773 + 0.634i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ 0.773 + 0.634i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.517539649\)
\(L(\frac12)\) \(\approx\) \(1.517539649\)
\(L(1)\) 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 \( 1 + (-0.923 - 0.382i)T \)
good3 \( 1 + (-0.923 + 0.382i)T^{2} \)
5 \( 1 + (0.382 - 0.923i)T^{2} \)
11 \( 1 + (-0.324 + 1.63i)T + (-0.923 - 0.382i)T^{2} \)
13 \( 1 + (0.382 + 0.923i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (-0.382 - 0.923i)T^{2} \)
23 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.216 + 1.08i)T + (-0.923 + 0.382i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.324 + 0.216i)T + (0.382 - 0.923i)T^{2} \)
41 \( 1 + (0.707 - 0.707i)T^{2} \)
43 \( 1 + (0.382 + 0.0761i)T + (0.923 + 0.382i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (0.382 - 1.92i)T + (-0.923 - 0.382i)T^{2} \)
59 \( 1 + (0.382 - 0.923i)T^{2} \)
61 \( 1 + (-0.923 + 0.382i)T^{2} \)
67 \( 1 + (1.92 - 0.382i)T + (0.923 - 0.382i)T^{2} \)
71 \( 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
83 \( 1 + (-0.382 - 0.923i)T^{2} \)
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

−8.707183207599352653774465871315, −7.936552484361170372300379940896, −7.31727061619386046049601293630, −6.13629720267749325030479477039, −5.93218344298020509401337478245, −4.71717721193049108679084847448, −4.11781894814344171264688911636, −3.15710681451149678206240784836, −2.06052663825748978372921775620, −0.973386327739859013508760469825, 1.58594520824121668992304161100, 1.94467740070951014045742488902, 3.49157427010735364500772090721, 4.43585611507381204077067209030, 4.76117926766562735303858696327, 5.74363744227951602927858982211, 6.85778038971462878363132149714, 7.38718188423943261764261875682, 7.86637571405956608816485922922, 8.772418727159541515055104292906

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