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Properties

Label 2-3584-3584.1189-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  + (0.740 − 0.671i)2-s + (0.0980 − 0.995i)4-s + (−0.989 + 0.146i)7-s + (−0.595 − 0.803i)8-s + (0.857 + 0.514i)9-s + (−0.620 − 1.23i)11-s + (−0.634 + 0.773i)14-s + (−0.980 − 0.195i)16-s + (0.980 − 0.195i)18-s + (−1.28 − 0.496i)22-s + (−1.43 − 1.30i)23-s + (0.941 − 0.336i)25-s + (0.0490 + 0.998i)28-s + (−0.752 − 0.872i)29-s + (−0.857 + 0.514i)32-s + ⋯
L(s)  = 1  + (0.740 − 0.671i)2-s + (0.0980 − 0.995i)4-s + (−0.989 + 0.146i)7-s + (−0.595 − 0.803i)8-s + (0.857 + 0.514i)9-s + (−0.620 − 1.23i)11-s + (−0.634 + 0.773i)14-s + (−0.980 − 0.195i)16-s + (0.980 − 0.195i)18-s + (−1.28 − 0.496i)22-s + (−1.43 − 1.30i)23-s + (0.941 − 0.336i)25-s + (0.0490 + 0.998i)28-s + (−0.752 − 0.872i)29-s + (−0.857 + 0.514i)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: $\chi_{3584} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ -0.824 + 0.565i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.403432180\)
\(L(\frac12)\) \(\approx\) \(1.403432180\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.740 + 0.671i)T \)
7 \( 1 + (0.989 - 0.146i)T \)
good3 \( 1 + (-0.857 - 0.514i)T^{2} \)
5 \( 1 + (-0.941 + 0.336i)T^{2} \)
11 \( 1 + (0.620 + 1.23i)T + (-0.595 + 0.803i)T^{2} \)
13 \( 1 + (-0.903 + 0.427i)T^{2} \)
17 \( 1 + (0.980 - 0.195i)T^{2} \)
19 \( 1 + (0.0490 + 0.998i)T^{2} \)
23 \( 1 + (1.43 + 1.30i)T + (0.0980 + 0.995i)T^{2} \)
29 \( 1 + (0.752 + 0.872i)T + (-0.146 + 0.989i)T^{2} \)
31 \( 1 + (0.923 + 0.382i)T^{2} \)
37 \( 1 + (-0.364 + 0.822i)T + (-0.671 - 0.740i)T^{2} \)
41 \( 1 + (-0.773 - 0.634i)T^{2} \)
43 \( 1 + (-0.983 + 1.73i)T + (-0.514 - 0.857i)T^{2} \)
47 \( 1 + (0.831 + 0.555i)T^{2} \)
53 \( 1 + (0.803 - 0.931i)T + (-0.146 - 0.989i)T^{2} \)
59 \( 1 + (0.903 + 0.427i)T^{2} \)
61 \( 1 + (0.970 - 0.242i)T^{2} \)
67 \( 1 + (1.07 - 1.37i)T + (-0.242 - 0.970i)T^{2} \)
71 \( 1 + (-0.914 - 0.229i)T + (0.881 + 0.471i)T^{2} \)
73 \( 1 + (-0.956 - 0.290i)T^{2} \)
79 \( 1 + (-0.754 - 0.403i)T + (0.555 + 0.831i)T^{2} \)
83 \( 1 + (0.671 - 0.740i)T^{2} \)
89 \( 1 + (-0.0980 + 0.995i)T^{2} \)
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

−8.568758219245914492251200777517, −7.62060830013646198166199608954, −6.72640139993524369043589387618, −5.99545358383913279023668352596, −5.48469056304567593929908205713, −4.37464913020947201139087585464, −3.81933466349013604074741832651, −2.80016519718211108605145600588, −2.16279811341262010328034280951, −0.60489842773838596858498628741, 1.77903614187344404560144969292, 2.99092135041005704566375104627, 3.71157901098220770529105253849, 4.49845837235278371690555478865, 5.22978958278363585924128726001, 6.20792534053969398783429256419, 6.70953704535269656497988891375, 7.51230649104716778661418120347, 7.84205147779081757456057901904, 9.161439271960755166553159178913

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