Properties

Label 2-1008-7.3-c2-0-13
Degree $2$
Conductor $1008$
Sign $0.124 - 0.992i$
Analytic cond. $27.4660$
Root an. cond. $5.24080$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.24 − 0.717i)5-s + (−1.74 + 6.77i)7-s + (−3 + 5.19i)11-s − 21.3i·13-s + (7.75 + 4.47i)17-s + (6.25 − 3.61i)19-s + (18.7 + 32.4i)23-s + (−11.4 + 19.8i)25-s + 33.9·29-s + (−38.2 − 22.0i)31-s + (2.69 + 9.67i)35-s + (13.9 + 24.2i)37-s + 54.8i·41-s + 1.48·43-s + (−37.2 + 21.5i)47-s + ⋯
L(s)  = 1  + (0.248 − 0.143i)5-s + (−0.248 + 0.968i)7-s + (−0.272 + 0.472i)11-s − 1.64i·13-s + (0.456 + 0.263i)17-s + (0.329 − 0.190i)19-s + (0.814 + 1.41i)23-s + (−0.458 + 0.794i)25-s + 1.17·29-s + (−1.23 − 0.711i)31-s + (0.0770 + 0.276i)35-s + (0.377 + 0.654i)37-s + 1.33i·41-s + 0.0345·43-s + (−0.792 + 0.457i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.124 - 0.992i$
Analytic conductor: \(27.4660\)
Root analytic conductor: \(5.24080\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1),\ 0.124 - 0.992i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.566710130\)
\(L(\frac12)\) \(\approx\) \(1.566710130\)
\(L(2)\) 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 \( 1 \)
7 \( 1 + (1.74 - 6.77i)T \)
good5 \( 1 + (-1.24 + 0.717i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 21.3iT - 169T^{2} \)
17 \( 1 + (-7.75 - 4.47i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-6.25 + 3.61i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-18.7 - 32.4i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 33.9T + 841T^{2} \)
31 \( 1 + (38.2 + 22.0i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-13.9 - 24.2i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 54.8iT - 1.68e3T^{2} \)
43 \( 1 - 1.48T + 1.84e3T^{2} \)
47 \( 1 + (37.2 - 21.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (42.7 - 74.0i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (35.6 + 20.6i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (1.02 - 0.594i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-2.19 + 3.80i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 137.T + 5.04e3T^{2} \)
73 \( 1 + (-68.3 - 39.4i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-49.1 - 85.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 110. iT - 6.88e3T^{2} \)
89 \( 1 + (-18 + 10.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 10.9iT - 9.40e3T^{2} \)
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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

−9.638186712528875468037703205196, −9.455933029991530835822531855576, −8.141773207249430810227744841137, −7.68714592471352877051251636115, −6.44070520269187062915379519426, −5.50414955676781437701264807342, −5.06742402666322240075023573700, −3.45425345903990108937269378947, −2.67756918131815117072728201818, −1.26998345302792874959749457181, 0.52302057431426142969610027722, 1.96350184430702811453529542267, 3.26171670497122543493099517573, 4.23320110082342686120197264991, 5.14135122716474241672923514819, 6.44295562451517239081153280380, 6.86748361962607562041309285228, 7.87540069067025351407653819605, 8.836020497594588575485366897634, 9.598700227070309132933828342403

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