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Properties

Label 2-1584-33.17-c1-0-19
Degree $2$
Conductor $1584$
Sign $-0.532 + 0.846i$
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  + (0.0381 + 0.0123i)5-s + (−0.145 − 0.199i)7-s + (−3.12 + 1.12i)11-s + (−2.18 + 0.711i)13-s + (−1.32 + 4.06i)17-s + (3.64 − 5.01i)19-s − 6.79i·23-s + (−4.04 − 2.93i)25-s + (4.52 − 3.28i)29-s + (−1.48 − 4.56i)31-s + (−0.00305 − 0.00941i)35-s + (3.26 − 2.36i)37-s + (−7.76 − 5.64i)41-s − 1.03i·43-s + (−6.53 + 9.00i)47-s + ⋯
L(s)  = 1  + (0.0170 + 0.00554i)5-s + (−0.0548 − 0.0754i)7-s + (−0.940 + 0.338i)11-s + (−0.607 + 0.197i)13-s + (−0.320 + 0.986i)17-s + (0.836 − 1.15i)19-s − 1.41i·23-s + (−0.808 − 0.587i)25-s + (0.840 − 0.610i)29-s + (−0.266 − 0.819i)31-s + (−0.000517 − 0.00159i)35-s + (0.536 − 0.389i)37-s + (−1.21 − 0.881i)41-s − 0.157i·43-s + (−0.953 + 1.31i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.532 + 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.532 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1584\)    =    \(2^{4} \cdot 3^{2} \cdot 11\)
Sign: $-0.532 + 0.846i$
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: \((2,\ 1584,\ (\ :1/2),\ -0.532 + 0.846i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7426119209\)
\(L(\frac12)\) \(\approx\) \(0.7426119209\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.12 - 1.12i)T \)
good5 \( 1 + (-0.0381 - 0.0123i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (0.145 + 0.199i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (2.18 - 0.711i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.32 - 4.06i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-3.64 + 5.01i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + 6.79iT - 23T^{2} \)
29 \( 1 + (-4.52 + 3.28i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.48 + 4.56i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-3.26 + 2.36i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (7.76 + 5.64i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 1.03iT - 43T^{2} \)
47 \( 1 + (6.53 - 9.00i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (8.52 - 2.77i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-1.63 - 2.25i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-8.06 - 2.62i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + 7.94T + 67T^{2} \)
71 \( 1 + (3.16 + 1.02i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (6.96 + 9.58i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.86 + 0.930i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.63 + 5.03i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 8.54iT - 89T^{2} \)
97 \( 1 + (0.935 + 2.88i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.171423369854665882679301210325, −8.285901466999243759616364418541, −7.61146573795854611154739968183, −6.74585218011673530849133506175, −5.92553690627055318368425010084, −4.88022954250498836805362468833, −4.24558814354571927930879321021, −2.89465943018785101554889152058, −2.07446857385783402250954042993, −0.27821443920307438751466613620, 1.48266930347799170898528109585, 2.83913427685762168592946091150, 3.55704457439739534165286946515, 5.02221327475651638947756124686, 5.36294954265979440654150601590, 6.48011899566414885987680377771, 7.47516388755836840963629621039, 7.937652702212028371388869976860, 8.923550241431101375735275289467, 9.862463436925764609462469041889

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