Properties

Label 2-1008-7.2-c1-0-15
Degree $2$
Conductor $1008$
Sign $-0.266 + 0.963i$
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  + (−0.5 + 0.866i)5-s + (2 − 1.73i)7-s + (−1.5 − 2.59i)11-s − 6·13-s + (−2.5 − 4.33i)17-s + (0.5 − 0.866i)19-s + (3.5 − 6.06i)23-s + (2 + 3.46i)25-s − 2·29-s + (−2.5 − 4.33i)31-s + (0.499 + 2.59i)35-s + (−1.5 + 2.59i)37-s + 2·41-s + 4·43-s + (−2.5 + 4.33i)47-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (0.755 − 0.654i)7-s + (−0.452 − 0.783i)11-s − 1.66·13-s + (−0.606 − 1.05i)17-s + (0.114 − 0.198i)19-s + (0.729 − 1.26i)23-s + (0.400 + 0.692i)25-s − 0.371·29-s + (−0.449 − 0.777i)31-s + (0.0845 + 0.439i)35-s + (−0.246 + 0.427i)37-s + 0.312·41-s + 0.609·43-s + (−0.364 + 0.631i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.266 + 0.963i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.028006477\)
\(L(\frac12)\) \(\approx\) \(1.028006477\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + 1.73i)T \)
good5 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
17 \( 1 + (2.5 + 4.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.5 + 6.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (2.5 - 4.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.5 + 12.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.5 + 7.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (3.5 + 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-3.5 + 6.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−9.710601953150148855921745639134, −8.945264058360348999736912160310, −7.81259332042344261872049267929, −7.36538705080999696288330089808, −6.48883265786195081405065911330, −5.08722790195235997598837495247, −4.65339369398508270856883753113, −3.25205905068220530516479109764, −2.28650678849815266617179821225, −0.45145050942181522872673004287, 1.70523669499749284534433395274, 2.69182966712179297451851881764, 4.22951617286718871558987262583, 4.98919013874357958876284647076, 5.69051225049821569371642582372, 7.08972686251183999837737529079, 7.65560837761174788327332283930, 8.593158643428850140872686802970, 9.293457820324809242173529516567, 10.19798157782018873181031389612

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