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Properties

Label 2-1008-28.23-c2-0-24
Degree $2$
Conductor $1008$
Sign $0.250 + 0.968i$
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  + (−3.04 + 5.26i)5-s + (2.29 − 6.61i)7-s + (−13.9 + 8.04i)11-s + 14·13-s + (−6.08 − 10.5i)17-s + (−4.58 − 2.64i)19-s + (−27.8 − 16.0i)23-s + (−6 − 10.3i)25-s + 42.5·29-s + (20.6 − 11.9i)31-s + (27.8 + 32.1i)35-s + (−19 + 32.9i)37-s − 24.3·41-s − 74.0i·43-s + (−38.5 − 30.3i)49-s + ⋯
L(s)  = 1  + (−0.608 + 1.05i)5-s + (0.327 − 0.944i)7-s + (−1.26 + 0.731i)11-s + 1.07·13-s + (−0.357 − 0.619i)17-s + (−0.241 − 0.139i)19-s + (−1.21 − 0.699i)23-s + (−0.239 − 0.415i)25-s + 1.46·29-s + (0.665 − 0.384i)31-s + (0.796 + 0.919i)35-s + (−0.513 + 0.889i)37-s − 0.593·41-s − 1.72i·43-s + (−0.785 − 0.618i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.250 + 0.968i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.250 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.250 + 0.968i$
Analytic conductor: \(27.4660\)
Root analytic conductor: \(5.24080\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1),\ 0.250 + 0.968i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.058414468\)
\(L(\frac12)\) \(\approx\) \(1.058414468\)
\(L(2)\) 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 \( 1 + (-2.29 + 6.61i)T \)
good5 \( 1 + (3.04 - 5.26i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (13.9 - 8.04i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 14T + 169T^{2} \)
17 \( 1 + (6.08 + 10.5i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (4.58 + 2.64i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (27.8 + 16.0i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 42.5T + 841T^{2} \)
31 \( 1 + (-20.6 + 11.9i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (19 - 32.9i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 24.3T + 1.68e3T^{2} \)
43 \( 1 + 74.0iT - 1.84e3T^{2} \)
47 \( 1 + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (21.2 + 36.8i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-97.5 + 56.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-96.2 + 55.5i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 64.3iT - 5.04e3T^{2} \)
73 \( 1 + (-7 - 12.1i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (48.1 + 27.7i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 112. iT - 6.88e3T^{2} \)
89 \( 1 + (-66.9 + 115. i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 35T + 9.40e3T^{2} \)
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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

−10.04005501545889787370396553727, −8.428984459498076824703401385250, −7.966850804809102195304146289694, −6.98964578779768254674622261588, −6.52953023267606682352983673707, −5.09444138517933045639815224380, −4.20604671494607634376175620832, −3.26245686815758682467685982173, −2.15277323445666933722668235695, −0.37112899487998109896254800148, 1.12983099942067893053491458172, 2.51604287462831889780095290243, 3.76261232009564031272062349862, 4.76477060389022892880392826025, 5.59743036171589354218087697158, 6.31685108552146454015353143351, 7.85153094769542114708913792234, 8.458761042911547324166043570348, 8.660751482589139630367743776284, 9.948788429976281889318990050512

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