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Properties

Label 2-3584-3584.1133-c0-0-0
Degree $2$
Conductor $3584$
Sign $0.824 + 0.565i$
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  + (0.998 − 0.0490i)2-s + (0.995 − 0.0980i)4-s + (0.803 − 0.595i)7-s + (0.989 − 0.146i)8-s + (−0.970 − 0.242i)9-s + (−0.717 − 0.0529i)11-s + (0.773 − 0.634i)14-s + (0.980 − 0.195i)16-s + (−0.980 − 0.195i)18-s + (−0.719 − 0.0176i)22-s + (1.02 + 0.0504i)23-s + (0.903 − 0.427i)25-s + (0.740 − 0.671i)28-s + (−0.759 − 1.50i)29-s + (0.970 − 0.242i)32-s + ⋯
L(s)  = 1  + (0.998 − 0.0490i)2-s + (0.995 − 0.0980i)4-s + (0.803 − 0.595i)7-s + (0.989 − 0.146i)8-s + (−0.970 − 0.242i)9-s + (−0.717 − 0.0529i)11-s + (0.773 − 0.634i)14-s + (0.980 − 0.195i)16-s + (−0.980 − 0.195i)18-s + (−0.719 − 0.0176i)22-s + (1.02 + 0.0504i)23-s + (0.903 − 0.427i)25-s + (0.740 − 0.671i)28-s + (−0.759 − 1.50i)29-s + (0.970 − 0.242i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $0.824 + 0.565i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (1133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ 0.824 + 0.565i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.517797536\)
\(L(\frac12)\) \(\approx\) \(2.517797536\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.998 + 0.0490i)T \)
7 \( 1 + (-0.803 + 0.595i)T \)
good3 \( 1 + (0.970 + 0.242i)T^{2} \)
5 \( 1 + (-0.903 + 0.427i)T^{2} \)
11 \( 1 + (0.717 + 0.0529i)T + (0.989 + 0.146i)T^{2} \)
13 \( 1 + (-0.336 - 0.941i)T^{2} \)
17 \( 1 + (-0.980 - 0.195i)T^{2} \)
19 \( 1 + (0.740 - 0.671i)T^{2} \)
23 \( 1 + (-1.02 - 0.0504i)T + (0.995 + 0.0980i)T^{2} \)
29 \( 1 + (0.759 + 1.50i)T + (-0.595 + 0.803i)T^{2} \)
31 \( 1 + (0.923 - 0.382i)T^{2} \)
37 \( 1 + (-1.44 - 1.37i)T + (0.0490 + 0.998i)T^{2} \)
41 \( 1 + (-0.634 - 0.773i)T^{2} \)
43 \( 1 + (-0.385 + 0.494i)T + (-0.242 - 0.970i)T^{2} \)
47 \( 1 + (-0.831 + 0.555i)T^{2} \)
53 \( 1 + (0.866 - 1.72i)T + (-0.595 - 0.803i)T^{2} \)
59 \( 1 + (0.336 - 0.941i)T^{2} \)
61 \( 1 + (-0.514 - 0.857i)T^{2} \)
67 \( 1 + (1.51 - 0.420i)T + (0.857 - 0.514i)T^{2} \)
71 \( 1 + (0.906 - 1.51i)T + (-0.471 - 0.881i)T^{2} \)
73 \( 1 + (-0.290 - 0.956i)T^{2} \)
79 \( 1 + (0.887 + 1.66i)T + (-0.555 + 0.831i)T^{2} \)
83 \( 1 + (-0.0490 + 0.998i)T^{2} \)
89 \( 1 + (-0.995 + 0.0980i)T^{2} \)
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

−8.433307658415787013539337996551, −7.79588833143851023445077559561, −7.17362942533015291031331630146, −6.20245712073844493865875240842, −5.62190663240326594853816041286, −4.76277199336691063116754634402, −4.22921588184266382915850600340, −3.06886400185710874639370545352, −2.50818494361811604330437529710, −1.15936408234033162518537914849, 1.59169855723400224898231291913, 2.63711632926097752188173138893, 3.15668106063081085696946618578, 4.38091535978411630387664638795, 5.25090004965338780577223479782, 5.43518548766281986349130839883, 6.40097729548261147392332420115, 7.33563369368963794398714688319, 7.907813644948070121359727026718, 8.692404650876399810239367305829

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