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Properties

Label 2-1008-21.17-c1-0-6
Degree $2$
Conductor $1008$
Sign $0.856 - 0.515i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.144 + 0.250i)5-s + (2.26 + 1.36i)7-s + (5.23 + 3.01i)11-s − 5.46i·13-s + (−2.22 + 3.85i)17-s + (−3.51 + 2.03i)19-s + (−1.11 + 0.645i)23-s + (2.45 − 4.25i)25-s + 0.377i·29-s + (3.09 + 1.78i)31-s + (−0.0126 + 0.766i)35-s + (1.01 + 1.75i)37-s − 5.50·41-s + 6.45·43-s + (5.38 + 9.33i)47-s + ⋯
L(s)  = 1  + (0.0647 + 0.112i)5-s + (0.857 + 0.514i)7-s + (1.57 + 0.910i)11-s − 1.51i·13-s + (−0.540 + 0.935i)17-s + (−0.806 + 0.465i)19-s + (−0.232 + 0.134i)23-s + (0.491 − 0.851i)25-s + 0.0701i·29-s + (0.556 + 0.321i)31-s + (−0.00214 + 0.129i)35-s + (0.166 + 0.289i)37-s − 0.859·41-s + 0.984·43-s + (0.785 + 1.36i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.856 - 0.515i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.856 - 0.515i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.886511252\)
\(L(\frac12)\) \(\approx\) \(1.886511252\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.26 - 1.36i)T \)
good5 \( 1 + (-0.144 - 0.250i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.23 - 3.01i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.46iT - 13T^{2} \)
17 \( 1 + (2.22 - 3.85i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.51 - 2.03i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.11 - 0.645i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.377iT - 29T^{2} \)
31 \( 1 + (-3.09 - 1.78i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.01 - 1.75i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.50T + 41T^{2} \)
43 \( 1 - 6.45T + 43T^{2} \)
47 \( 1 + (-5.38 - 9.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-9.77 - 5.64i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.790 + 1.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.54 + 5.51i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.04 + 3.53i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.410iT - 71T^{2} \)
73 \( 1 + (11.1 + 6.41i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.01 + 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.155T + 83T^{2} \)
89 \( 1 + (3.34 + 5.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.5iT - 97T^{2} \)
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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

−10.19550602430202047221926426163, −9.060348511290153417995580372152, −8.435857892987952104631551552454, −7.64134073135395835314606650297, −6.53287013524063808498100961108, −5.85289405067366841466100799909, −4.69176102170339663095304282627, −3.92783863618947671791831707977, −2.49634489334945319146422458543, −1.37841145466071230861970633191, 1.03982138388953356249835519303, 2.25522239316860809256872679945, 3.90021718316997922841077348849, 4.40000909466979929303877835069, 5.56413023147083539497368613321, 6.78427087846305509474893655997, 7.06394812668417840944721013242, 8.606518852041868369519100928168, 8.827272999669037522367818808642, 9.785900955776882007535496620028

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