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Properties

Label 2-1008-252.187-c1-0-17
Degree $2$
Conductor $1008$
Sign $-0.539 - 0.842i$
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  + (0.840 + 1.51i)3-s + 2.34i·5-s + (2.61 + 0.373i)7-s + (−1.58 + 2.54i)9-s + 0.442i·11-s + (−1.96 + 1.13i)13-s + (−3.55 + 1.97i)15-s + (4.01 − 2.32i)17-s + (−3.30 + 5.73i)19-s + (1.63 + 4.28i)21-s − 1.13i·23-s − 0.514·25-s + (−5.18 − 0.267i)27-s + (4.37 − 7.58i)29-s + (−2.32 + 4.01i)31-s + ⋯
L(s)  = 1  + (0.485 + 0.874i)3-s + 1.05i·5-s + (0.989 + 0.141i)7-s + (−0.529 + 0.848i)9-s + 0.133i·11-s + (−0.543 + 0.314i)13-s + (−0.918 + 0.509i)15-s + (0.974 − 0.562i)17-s + (−0.759 + 1.31i)19-s + (0.356 + 0.934i)21-s − 0.236i·23-s − 0.102·25-s + (−0.998 − 0.0515i)27-s + (0.812 − 1.40i)29-s + (−0.416 + 0.721i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.539 - 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.539 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.539 - 0.842i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.539 - 0.842i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.920895874\)
\(L(\frac12)\) \(\approx\) \(1.920895874\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.840 - 1.51i)T \)
7 \( 1 + (-2.61 - 0.373i)T \)
good5 \( 1 - 2.34iT - 5T^{2} \)
11 \( 1 - 0.442iT - 11T^{2} \)
13 \( 1 + (1.96 - 1.13i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-4.01 + 2.32i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.30 - 5.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.13iT - 23T^{2} \)
29 \( 1 + (-4.37 + 7.58i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.32 - 4.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.41 - 2.44i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.55 + 2.62i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.81 + 2.20i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.67 + 4.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.54 + 2.67i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.689 - 1.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.88 - 5.13i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.10 - 1.21i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.20iT - 71T^{2} \)
73 \( 1 + (-3.17 + 1.83i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.26 + 4.19i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.87 - 13.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.32 - 3.07i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-14.9 - 8.63i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.26482060265230880774690007999, −9.615620031152461354943804477765, −8.493133870226677635929990563177, −7.905879943679697124542325822206, −7.01933283097486883172921652261, −5.84588792245026642328206651753, −4.91000890442890078047368194383, −4.01151164497447863732028963719, −2.96967904113476592325491842528, −2.00699989782905030023107826272, 0.856887634041106552212204012146, 1.87109750159843865833544747191, 3.15067882167686014764604458363, 4.52737822673152156949310466722, 5.24957884952684272222528940887, 6.33159900830591290745743962870, 7.42980299495439877718824724409, 8.015881922716764302385312502744, 8.747040027341838490365993310078, 9.340011702950142830336197411289

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