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Properties

Label 2-3584-224.181-c0-0-3
Degree $2$
Conductor $3584$
Sign $-0.980 + 0.195i$
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  + (−0.707 − 0.707i)7-s + (0.707 − 0.707i)9-s + (−1.70 − 0.707i)11-s + (−1 + i)23-s + (−0.707 − 0.707i)25-s + (−1.70 + 0.707i)29-s + (−0.292 + 0.707i)37-s + (0.707 + 0.292i)43-s + 1.00i·49-s + (−0.707 − 0.292i)53-s − 1.00·63-s + (−0.707 + 0.292i)67-s + (0.707 + 1.70i)77-s − 1.41i·79-s − 1.00i·81-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)7-s + (0.707 − 0.707i)9-s + (−1.70 − 0.707i)11-s + (−1 + i)23-s + (−0.707 − 0.707i)25-s + (−1.70 + 0.707i)29-s + (−0.292 + 0.707i)37-s + (0.707 + 0.292i)43-s + 1.00i·49-s + (−0.707 − 0.292i)53-s − 1.00·63-s + (−0.707 + 0.292i)67-s + (0.707 + 1.70i)77-s − 1.41i·79-s − 1.00i·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $-0.980 + 0.195i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (1217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ -0.980 + 0.195i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3503615694\)
\(L(\frac12)\) \(\approx\) \(0.3503615694\)
\(L(1)\) 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 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-0.707 + 0.707i)T^{2} \)
5 \( 1 + (0.707 + 0.707i)T^{2} \)
11 \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \)
13 \( 1 + (0.707 - 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (0.707 - 0.707i)T^{2} \)
23 \( 1 + (1 - i)T - iT^{2} \)
29 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.707 + 0.707i)T^{2} \)
61 \( 1 + (-0.707 + 0.707i)T^{2} \)
67 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 + (0.707 - 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
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

−8.215767745289149720296122265342, −7.63134942120608359706511449029, −7.02446133062925967446556932691, −6.06486796737626047108309702127, −5.54462514033664121018387147239, −4.43590779421409828352644563031, −3.63835955895356237986624984866, −2.97399246043269776412029238366, −1.68856494803494120002271164551, −0.18489734317147735298498070942, 2.03792795615319591688982556547, 2.44864436639416112236042640109, 3.66918454437157737967395871898, 4.56488312573743181669197896855, 5.41928835519669893166588448630, 5.92006208450241034188729142527, 7.00980166210813397258227083386, 7.67435035841717016211406010833, 8.135609227493713497694715813018, 9.229118675015946207389824098017

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