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Properties

Label 2-3584-3584.2197-c0-0-0
Degree $2$
Conductor $3584$
Sign $0.876 - 0.482i$
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  + (−0.242 + 0.970i)2-s + (−0.881 − 0.471i)4-s + (0.0490 − 0.998i)7-s + (0.671 − 0.740i)8-s + (0.941 + 0.336i)9-s + (1.59 + 0.705i)11-s + (0.956 + 0.290i)14-s + (0.555 + 0.831i)16-s + (−0.555 + 0.831i)18-s + (−1.07 + 1.37i)22-s + (−0.439 − 1.75i)23-s + (−0.803 − 0.595i)25-s + (−0.514 + 0.857i)28-s + (−0.898 + 0.0220i)29-s + (−0.941 + 0.336i)32-s + ⋯
L(s)  = 1  + (−0.242 + 0.970i)2-s + (−0.881 − 0.471i)4-s + (0.0490 − 0.998i)7-s + (0.671 − 0.740i)8-s + (0.941 + 0.336i)9-s + (1.59 + 0.705i)11-s + (0.956 + 0.290i)14-s + (0.555 + 0.831i)16-s + (−0.555 + 0.831i)18-s + (−1.07 + 1.37i)22-s + (−0.439 − 1.75i)23-s + (−0.803 − 0.595i)25-s + (−0.514 + 0.857i)28-s + (−0.898 + 0.0220i)29-s + (−0.941 + 0.336i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $0.876 - 0.482i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (2197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ 0.876 - 0.482i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.211524936\)
\(L(\frac12)\) \(\approx\) \(1.211524936\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.242 - 0.970i)T \)
7 \( 1 + (-0.0490 + 0.998i)T \)
good3 \( 1 + (-0.941 - 0.336i)T^{2} \)
5 \( 1 + (0.803 + 0.595i)T^{2} \)
11 \( 1 + (-1.59 - 0.705i)T + (0.671 + 0.740i)T^{2} \)
13 \( 1 + (-0.146 + 0.989i)T^{2} \)
17 \( 1 + (-0.555 + 0.831i)T^{2} \)
19 \( 1 + (-0.514 + 0.857i)T^{2} \)
23 \( 1 + (0.439 + 1.75i)T + (-0.881 + 0.471i)T^{2} \)
29 \( 1 + (0.898 - 0.0220i)T + (0.998 - 0.0490i)T^{2} \)
31 \( 1 + (-0.382 + 0.923i)T^{2} \)
37 \( 1 + (-1.81 - 0.223i)T + (0.970 + 0.242i)T^{2} \)
41 \( 1 + (-0.290 + 0.956i)T^{2} \)
43 \( 1 + (-0.794 + 1.12i)T + (-0.336 - 0.941i)T^{2} \)
47 \( 1 + (0.980 - 0.195i)T^{2} \)
53 \( 1 + (1.68 + 0.0414i)T + (0.998 + 0.0490i)T^{2} \)
59 \( 1 + (0.146 + 0.989i)T^{2} \)
61 \( 1 + (-0.903 + 0.427i)T^{2} \)
67 \( 1 + (-1.38 - 0.874i)T + (0.427 + 0.903i)T^{2} \)
71 \( 1 + (1.39 + 0.661i)T + (0.634 + 0.773i)T^{2} \)
73 \( 1 + (0.995 - 0.0980i)T^{2} \)
79 \( 1 + (-1.25 - 1.52i)T + (-0.195 + 0.980i)T^{2} \)
83 \( 1 + (-0.970 + 0.242i)T^{2} \)
89 \( 1 + (0.881 + 0.471i)T^{2} \)
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

−8.700492911180586191199476164402, −7.81426540746663211786988603999, −7.35600312079311423841673210005, −6.57256799382774833953622964645, −6.20181580753575044684827304119, −4.88066736288583761691716151449, −4.19132966264723645856995634071, −3.92277800554719158664573358971, −2.00511688225706705849064862643, −0.956649381620247521302332064655, 1.26380501915958051082700266057, 1.94625915926776088448374272783, 3.24277501136238969558582190548, 3.80864233187613693968513068688, 4.60598033784117269164889133922, 5.71967058262639880596500179923, 6.26614203187269889567291866049, 7.47441957622078025805169642132, 8.018099497401209383374054918981, 9.102278607059299699110725944127

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