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Properties

Label 2-3584-3584.1749-c0-0-0
Degree $2$
Conductor $3584$
Sign $0.960 + 0.278i$
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  + (−0.514 − 0.857i)2-s + (−0.471 + 0.881i)4-s + (−0.671 − 0.740i)7-s + (0.998 − 0.0490i)8-s + (0.427 + 0.903i)9-s + (0.244 + 0.00600i)11-s + (−0.290 + 0.956i)14-s + (−0.555 − 0.831i)16-s + (0.555 − 0.831i)18-s + (−0.120 − 0.213i)22-s + (−0.968 + 1.61i)23-s + (0.989 − 0.146i)25-s + (0.970 − 0.242i)28-s + (1.21 + 0.470i)29-s + (−0.427 + 0.903i)32-s + ⋯
L(s)  = 1  + (−0.514 − 0.857i)2-s + (−0.471 + 0.881i)4-s + (−0.671 − 0.740i)7-s + (0.998 − 0.0490i)8-s + (0.427 + 0.903i)9-s + (0.244 + 0.00600i)11-s + (−0.290 + 0.956i)14-s + (−0.555 − 0.831i)16-s + (0.555 − 0.831i)18-s + (−0.120 − 0.213i)22-s + (−0.968 + 1.61i)23-s + (0.989 − 0.146i)25-s + (0.970 − 0.242i)28-s + (1.21 + 0.470i)29-s + (−0.427 + 0.903i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $0.960 + 0.278i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (1749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ 0.960 + 0.278i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8621890965\)
\(L(\frac12)\) \(\approx\) \(0.8621890965\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.514 + 0.857i)T \)
7 \( 1 + (0.671 + 0.740i)T \)
good3 \( 1 + (-0.427 - 0.903i)T^{2} \)
5 \( 1 + (-0.989 + 0.146i)T^{2} \)
11 \( 1 + (-0.244 - 0.00600i)T + (0.998 + 0.0490i)T^{2} \)
13 \( 1 + (0.803 - 0.595i)T^{2} \)
17 \( 1 + (0.555 - 0.831i)T^{2} \)
19 \( 1 + (0.970 - 0.242i)T^{2} \)
23 \( 1 + (0.968 - 1.61i)T + (-0.471 - 0.881i)T^{2} \)
29 \( 1 + (-1.21 - 0.470i)T + (0.740 + 0.671i)T^{2} \)
31 \( 1 + (-0.382 + 0.923i)T^{2} \)
37 \( 1 + (0.533 + 1.92i)T + (-0.857 + 0.514i)T^{2} \)
41 \( 1 + (-0.956 - 0.290i)T^{2} \)
43 \( 1 + (0.400 - 1.78i)T + (-0.903 - 0.427i)T^{2} \)
47 \( 1 + (-0.980 + 0.195i)T^{2} \)
53 \( 1 + (-1.07 + 0.414i)T + (0.740 - 0.671i)T^{2} \)
59 \( 1 + (-0.803 - 0.595i)T^{2} \)
61 \( 1 + (-0.941 - 0.336i)T^{2} \)
67 \( 1 + (-1.12 - 1.59i)T + (-0.336 + 0.941i)T^{2} \)
71 \( 1 + (-1.19 + 0.427i)T + (0.773 - 0.634i)T^{2} \)
73 \( 1 + (0.0980 + 0.995i)T^{2} \)
79 \( 1 + (0.920 - 0.755i)T + (0.195 - 0.980i)T^{2} \)
83 \( 1 + (0.857 + 0.514i)T^{2} \)
89 \( 1 + (0.471 - 0.881i)T^{2} \)
97 \( 1 + (0.923 + 0.382i)T^{2} \)
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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

−8.771024180124253654379142521494, −8.051618231661806192203532776353, −7.32252175090875774017821035627, −6.80834769305913016195261623461, −5.58620572463570631539741407362, −4.62464257867064997593594172508, −3.91253788286225409901825160344, −3.13761005044419659363629807562, −2.10846692417493776576284124033, −1.07406748814019477394624681571, 0.74458963526957032012964186650, 2.15131971054200829215604934004, 3.32595842966911289119812680738, 4.34436897429337879293398683585, 5.12071034686983250794898991246, 6.14136461811609606882219475369, 6.53957482240985377589845015540, 7.06052274936137432353943139467, 8.272964432141850490408882302631, 8.654337721239497585058175428760

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