Properties

Label 2-1008-21.5-c3-0-47
Degree $2$
Conductor $1008$
Sign $-0.974 - 0.223i$
Analytic cond. $59.4739$
Root an. cond. $7.71193$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (10.9 − 18.9i)5-s + (12.1 − 13.9i)7-s + (−45.1 + 26.0i)11-s + 54.9i·13-s + (40.8 + 70.7i)17-s + (−113. − 65.6i)19-s + (−38.0 − 21.9i)23-s + (−176. − 305. i)25-s − 238. i·29-s + (−174. + 100. i)31-s + (−132. − 382. i)35-s + (12.0 − 20.8i)37-s − 102.·41-s − 119.·43-s + (−20.2 + 35.1i)47-s + ⋯
L(s)  = 1  + (0.977 − 1.69i)5-s + (0.655 − 0.755i)7-s + (−1.23 + 0.714i)11-s + 1.17i·13-s + (0.582 + 1.00i)17-s + (−1.37 − 0.792i)19-s + (−0.345 − 0.199i)23-s + (−1.40 − 2.44i)25-s − 1.52i·29-s + (−1.00 + 0.582i)31-s + (−0.638 − 1.84i)35-s + (0.0534 − 0.0925i)37-s − 0.389·41-s − 0.424·43-s + (−0.0629 + 0.109i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.223i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.974 - 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.974 - 0.223i$
Analytic conductor: \(59.4739\)
Root analytic conductor: \(7.71193\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :3/2),\ -0.974 - 0.223i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8571442308\)
\(L(\frac12)\) \(\approx\) \(0.8571442308\)
\(L(\frac{5}{2})\) 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 \( 1 + (-12.1 + 13.9i)T \)
good5 \( 1 + (-10.9 + 18.9i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (45.1 - 26.0i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 54.9iT - 2.19e3T^{2} \)
17 \( 1 + (-40.8 - 70.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (113. + 65.6i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (38.0 + 21.9i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 238. iT - 2.43e4T^{2} \)
31 \( 1 + (174. - 100. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-12.0 + 20.8i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 102.T + 6.89e4T^{2} \)
43 \( 1 + 119.T + 7.95e4T^{2} \)
47 \( 1 + (20.2 - 35.1i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (297. - 172. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-142. - 246. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (386. + 223. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (113. + 196. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 886. iT - 3.57e5T^{2} \)
73 \( 1 + (-6.46 + 3.73i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-404. + 700. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 943.T + 5.71e5T^{2} \)
89 \( 1 + (-575. + 996. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.55e3iT - 9.12e5T^{2} \)
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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

−9.077485634091827216777932272733, −8.385423692581241154206837475837, −7.67568765322004070884457523161, −6.46464571191673653505411082169, −5.52765350155575827264078640022, −4.62900103594807983985506559431, −4.24432150350662434561719860812, −2.14009208842895260185710380548, −1.57659588499947701906327146031, −0.18583064037201527500609855434, 1.84686421336419100796159012381, 2.73725142792428978558819762674, 3.36624337556458559689650459215, 5.28588317344705852453171311399, 5.63657283130946596005784063032, 6.54186468456659857255713894612, 7.60098289631384597926098884163, 8.192170915710788278417174994893, 9.336158594386227279222100961228, 10.27178079018893454260444052669

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