Properties

Label 2-1008-21.5-c3-0-13
Degree $2$
Conductor $1008$
Sign $0.635 - 0.772i$
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  + (7.29 − 12.6i)5-s + (−18.4 + 0.998i)7-s + (38.7 − 22.3i)11-s + 76.8i·13-s + (58.3 + 101. i)17-s + (−83.5 − 48.2i)19-s + (−6.88 − 3.97i)23-s + (−43.9 − 76.1i)25-s − 86.8i·29-s + (−216. + 124. i)31-s + (−122. + 240. i)35-s + (−160. + 277. i)37-s − 231.·41-s + 413.·43-s + (−235. + 407. i)47-s + ⋯
L(s)  = 1  + (0.652 − 1.13i)5-s + (−0.998 + 0.0539i)7-s + (1.06 − 0.612i)11-s + 1.63i·13-s + (0.832 + 1.44i)17-s + (−1.00 − 0.582i)19-s + (−0.0624 − 0.0360i)23-s + (−0.351 − 0.608i)25-s − 0.556i·29-s + (−1.25 + 0.723i)31-s + (−0.590 + 1.16i)35-s + (−0.711 + 1.23i)37-s − 0.881·41-s + 1.46·43-s + (−0.730 + 1.26i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 - 0.772i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.635 - 0.772i$
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: \((2,\ 1008,\ (\ :3/2),\ 0.635 - 0.772i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.732103783\)
\(L(\frac12)\) \(\approx\) \(1.732103783\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.4 - 0.998i)T \)
good5 \( 1 + (-7.29 + 12.6i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-38.7 + 22.3i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 76.8iT - 2.19e3T^{2} \)
17 \( 1 + (-58.3 - 101. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (83.5 + 48.2i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (6.88 + 3.97i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 86.8iT - 2.43e4T^{2} \)
31 \( 1 + (216. - 124. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (160. - 277. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 231.T + 6.89e4T^{2} \)
43 \( 1 - 413.T + 7.95e4T^{2} \)
47 \( 1 + (235. - 407. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-600. + 346. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (143. + 249. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-740. - 427. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (240. + 416. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 930. iT - 3.57e5T^{2} \)
73 \( 1 + (98.4 - 56.8i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (111. - 192. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 692.T + 5.71e5T^{2} \)
89 \( 1 + (258. - 448. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 807. iT - 9.12e5T^{2} \)
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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

−9.504036308677792771910337767065, −8.921866651833503516757279131832, −8.433550290997615357042740899423, −6.87181844991315365404721910315, −6.32120958445836804109726842201, −5.51141653266827023885333626930, −4.30959010740935672763449498676, −3.59922506233401240593484210315, −2.00264707604247369852767288991, −1.09871242526692407234510005202, 0.46717812332254721611561020737, 2.11174099717352902144820223947, 3.09741491512682472605526374403, 3.82765211091892161808522056188, 5.41673925925754790117514838423, 6.05840623985751058821497950442, 7.00624879534587923138825128368, 7.44934692693881747221181717681, 8.841885224637999105685361023825, 9.652962636434894202728155739287

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