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Properties

Label 2-1584-33.17-c1-0-3
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: $\chi_{1584} (17, \cdot )$
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

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

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