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Properties

Label 2-1008-21.2-c0-0-0
Degree $2$
Conductor $1008$
Sign $-0.851 - 0.524i$
Analytic cond. $0.503057$
Root an. cond. $0.709265$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 0.707i)5-s + (−0.5 + 0.866i)7-s + (−1.22 + 0.707i)11-s − 13-s + (−0.5 + 0.866i)19-s + (0.499 + 0.866i)25-s + (−0.5 − 0.866i)31-s + (1.22 − 0.707i)35-s + (−0.5 + 0.866i)37-s + 1.41i·41-s + 43-s + (−1.22 − 0.707i)47-s + (−0.499 − 0.866i)49-s + 2·55-s + (1.22 + 0.707i)65-s + ⋯
L(s)  = 1  + (−1.22 − 0.707i)5-s + (−0.5 + 0.866i)7-s + (−1.22 + 0.707i)11-s − 13-s + (−0.5 + 0.866i)19-s + (0.499 + 0.866i)25-s + (−0.5 − 0.866i)31-s + (1.22 − 0.707i)35-s + (−0.5 + 0.866i)37-s + 1.41i·41-s + 43-s + (−1.22 − 0.707i)47-s + (−0.499 − 0.866i)49-s + 2·55-s + (1.22 + 0.707i)65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.851 - 0.524i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :0),\ -0.851 - 0.524i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2010716655\)
\(L(\frac12)\) \(\approx\) \(0.2010716655\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−10.35669186959104996696296019564, −9.699129369716018495139620477280, −8.748819019362095814093471854051, −7.950646564031574696113835090835, −7.45994507190752733238767010073, −6.21996387356881684044387915527, −5.13567215735783345855231034502, −4.51154466921230235228734436905, −3.30163356021236317721278618921, −2.16399280714451659372404236865, 0.17505609406316021203943626829, 2.62804268978152513700220461899, 3.44798733023799191231462458514, 4.38463582948927529282666238910, 5.43690806510760131763004472120, 6.75326508655141925745908134009, 7.33139677408003243034124460075, 7.924015944353601385400201802547, 8.931203825246533758155290959458, 10.03503827289418994736538453508

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