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Properties

Label 2-1584-33.2-c1-0-17
Degree $2$
Conductor $1584$
Sign $-0.311 + 0.950i$
Analytic cond. $12.6483$
Root an. cond. $3.55644$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.13 + 0.694i)5-s + (2.38 − 3.27i)7-s + (−3.31 + 0.0200i)11-s + (4.42 + 1.43i)13-s + (0.0235 + 0.0725i)17-s + (−1.40 − 1.93i)19-s − 3.22i·23-s + (0.0437 − 0.0318i)25-s + (−1.48 − 1.08i)29-s + (−0.517 + 1.59i)31-s + (−2.81 + 8.66i)35-s + (−5.87 − 4.27i)37-s + (6.82 − 4.96i)41-s − 4.28i·43-s + (−3.65 − 5.02i)47-s + ⋯
L(s)  = 1  + (−0.956 + 0.310i)5-s + (0.899 − 1.23i)7-s + (−0.999 + 0.00604i)11-s + (1.22 + 0.398i)13-s + (0.00571 + 0.0175i)17-s + (−0.323 − 0.444i)19-s − 0.672i·23-s + (0.00875 − 0.00636i)25-s + (−0.276 − 0.200i)29-s + (−0.0929 + 0.286i)31-s + (−0.475 + 1.46i)35-s + (−0.966 − 0.702i)37-s + (1.06 − 0.774i)41-s − 0.652i·43-s + (−0.532 − 0.733i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.311 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1584\)    =    \(2^{4} \cdot 3^{2} \cdot 11\)
Sign: $-0.311 + 0.950i$
Analytic conductor: \(12.6483\)
Root analytic conductor: \(3.55644\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1584} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1584,\ (\ :1/2),\ -0.311 + 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.028363603\)
\(L(\frac12)\) \(\approx\) \(1.028363603\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 + (3.31 - 0.0200i)T \)
good5 \( 1 + (2.13 - 0.694i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (-2.38 + 3.27i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-4.42 - 1.43i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.0235 - 0.0725i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.40 + 1.93i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + 3.22iT - 23T^{2} \)
29 \( 1 + (1.48 + 1.08i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.517 - 1.59i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (5.87 + 4.27i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-6.82 + 4.96i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 4.28iT - 43T^{2} \)
47 \( 1 + (3.65 + 5.02i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.16 + 0.379i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (0.341 - 0.469i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (3.59 - 1.16i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 12.9T + 67T^{2} \)
71 \( 1 + (-1.06 + 0.346i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-7.82 + 10.7i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.627 + 0.203i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.15 + 9.71i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 6.58iT - 89T^{2} \)
97 \( 1 + (-5.08 + 15.6i)T + (-78.4 - 57.0i)T^{2} \)
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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

−8.946280532828568763901577498728, −8.285704693815272709675600959662, −7.51913336880653299627441310928, −7.08623619489038102986171432443, −5.94270016269985342986956758887, −4.79907187960954190714093683021, −4.10011018515185069111718163169, −3.34318321947515957805282753823, −1.86630841897984795633356955127, −0.41488667260551190028123847745, 1.43152713261054220401135482974, 2.68967234257158877776082094700, 3.73410445484652456189714791091, 4.75535280193549508741664189997, 5.50684429295724955288872761328, 6.24337328275528299059778018717, 7.66181140917107600170522580136, 8.100323254710413298531702959631, 8.608106852163082997594626063395, 9.502597217530116335629852783871

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