Properties

Label 2-1008-21.5-c3-0-8
Degree $2$
Conductor $1008$
Sign $0.269 - 0.962i$
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  + (−0.632 + 1.09i)5-s + (−13.4 − 12.7i)7-s + (−36.0 + 20.7i)11-s − 85.7i·13-s + (38.8 + 67.3i)17-s + (−42.1 − 24.3i)19-s + (−78.7 − 45.4i)23-s + (61.6 + 106. i)25-s − 151. i·29-s + (−76.3 + 44.0i)31-s + (22.4 − 6.60i)35-s + (−45.2 + 78.4i)37-s + 383.·41-s + 227.·43-s + (−69.5 + 120. i)47-s + ⋯
L(s)  = 1  + (−0.0566 + 0.0980i)5-s + (−0.723 − 0.690i)7-s + (−0.987 + 0.570i)11-s − 1.82i·13-s + (0.554 + 0.961i)17-s + (−0.509 − 0.293i)19-s + (−0.714 − 0.412i)23-s + (0.493 + 0.854i)25-s − 0.968i·29-s + (−0.442 + 0.255i)31-s + (0.108 − 0.0318i)35-s + (−0.201 + 0.348i)37-s + 1.46·41-s + 0.808·43-s + (−0.215 + 0.373i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.269 - 0.962i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.269 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.269 - 0.962i$
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: \((2,\ 1008,\ (\ :3/2),\ 0.269 - 0.962i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9004050409\)
\(L(\frac12)\) \(\approx\) \(0.9004050409\)
\(L(\frac{5}{2})\) 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 \( 1 + (13.4 + 12.7i)T \)
good5 \( 1 + (0.632 - 1.09i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (36.0 - 20.7i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 85.7iT - 2.19e3T^{2} \)
17 \( 1 + (-38.8 - 67.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (42.1 + 24.3i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (78.7 + 45.4i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 151. iT - 2.43e4T^{2} \)
31 \( 1 + (76.3 - 44.0i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (45.2 - 78.4i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 383.T + 6.89e4T^{2} \)
43 \( 1 - 227.T + 7.95e4T^{2} \)
47 \( 1 + (69.5 - 120. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (289. - 167. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-440. - 762. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-11.3 - 6.57i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (221. + 383. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 341. iT - 3.57e5T^{2} \)
73 \( 1 + (798. - 460. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (206. - 357. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 954.T + 5.71e5T^{2} \)
89 \( 1 + (14.8 - 25.7i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.19e3iT - 9.12e5T^{2} \)
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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

−10.13728228025226950068229222302, −8.927628215980418429930600990106, −7.78317905128341563194104158656, −7.54766636770972110017345916488, −6.25343021871853999188381275611, −5.56607178404116457371136188065, −4.42222192833572135603257165055, −3.39830969958374496880901488676, −2.50391256286616747750806630039, −0.869984835922590963280884899809, 0.27920985540581251978144196456, 1.97747986044552037952024774533, 2.93405284218845597819542890498, 4.04603520837387523207996056793, 5.13376807844234131800448585937, 5.99891300042698530155344745641, 6.81063140072252704857433724027, 7.75298875409576550475085320450, 8.751785959261820240768091140513, 9.320846262127625870773441931048

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