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Properties

Label 2-1008-21.17-c1-0-10
Degree $2$
Conductor $1008$
Sign $-0.0137 + 0.999i$
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  + (−0.587 − 1.01i)5-s + (−2.35 + 1.21i)7-s + (1.44 + 0.835i)11-s − 1.14i·13-s + (2.07 − 3.59i)17-s + (−1.83 + 1.05i)19-s + (4.22 − 2.43i)23-s + (1.81 − 3.13i)25-s − 8.32i·29-s + (−7.18 − 4.14i)31-s + (2.61 + 1.68i)35-s + (−2.19 − 3.81i)37-s + 2.67·41-s − 4.08·43-s + (−1.75 − 3.03i)47-s + ⋯
L(s)  = 1  + (−0.262 − 0.454i)5-s + (−0.889 + 0.457i)7-s + (0.436 + 0.251i)11-s − 0.317i·13-s + (0.503 − 0.872i)17-s + (−0.420 + 0.242i)19-s + (0.880 − 0.508i)23-s + (0.362 − 0.627i)25-s − 1.54i·29-s + (−1.28 − 0.744i)31-s + (0.441 + 0.284i)35-s + (−0.361 − 0.626i)37-s + 0.417·41-s − 0.623·43-s + (−0.255 − 0.442i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0137 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0137 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.0137 + 0.999i$
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: \((2,\ 1008,\ (\ :1/2),\ -0.0137 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.055980836\)
\(L(\frac12)\) \(\approx\) \(1.055980836\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.35 - 1.21i)T \)
good5 \( 1 + (0.587 + 1.01i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.44 - 0.835i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.14iT - 13T^{2} \)
17 \( 1 + (-2.07 + 3.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.83 - 1.05i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.22 + 2.43i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.32iT - 29T^{2} \)
31 \( 1 + (7.18 + 4.14i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.19 + 3.81i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.67T + 41T^{2} \)
43 \( 1 + 4.08T + 43T^{2} \)
47 \( 1 + (1.75 + 3.03i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.59 - 0.922i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.98 + 5.17i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (12.8 - 7.39i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.22 + 7.31i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.48iT - 71T^{2} \)
73 \( 1 + (-0.846 - 0.488i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.56 - 9.63i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + (-6.29 - 10.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.52iT - 97T^{2} \)
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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

−9.597571100730449148582390731591, −9.063474205066494371708269256724, −8.151167963057650313634862073660, −7.21924295960170796951474882445, −6.33292161508406367852986652883, −5.46718779084040070926152984053, −4.43667049052137721745760015032, −3.41121174817107106332987257932, −2.29214188657053800574393688304, −0.50490843230729265885093622813, 1.43141586911129919552210958920, 3.17392496294107060539130616532, 3.64869277029200706478185101107, 4.93924969892419756482418015234, 6.06171160662297346171028889341, 6.89744671018021273423015856425, 7.41563165939066602946049763075, 8.702444205200590499405052454455, 9.255952841481890841394796624294, 10.32147752395836232273498094756

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