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Properties

Label 2-1008-21.17-c1-0-1
Degree $2$
Conductor $1008$
Sign $-0.778 - 0.627i$
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  + (1.22 + 2.12i)5-s + (−0.5 + 2.59i)7-s + (−3.67 − 2.12i)11-s + 1.73i·13-s + (−2.44 + 4.24i)17-s + (−4.5 + 2.59i)19-s + (−0.499 + 0.866i)25-s − 8.48i·29-s + (1.5 + 0.866i)31-s + (−6.12 + 2.12i)35-s + (2.5 + 4.33i)37-s − 12.2·41-s + 11·43-s + (−1.22 − 2.12i)47-s + (−6.5 − 2.59i)49-s + ⋯
L(s)  = 1  + (0.547 + 0.948i)5-s + (−0.188 + 0.981i)7-s + (−1.10 − 0.639i)11-s + 0.480i·13-s + (−0.594 + 1.02i)17-s + (−1.03 + 0.596i)19-s + (−0.0999 + 0.173i)25-s − 1.57i·29-s + (0.269 + 0.155i)31-s + (−1.03 + 0.358i)35-s + (0.410 + 0.711i)37-s − 1.91·41-s + 1.67·43-s + (−0.178 − 0.309i)47-s + (−0.928 − 0.371i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.778 - 0.627i$
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: \((2,\ 1008,\ (\ :1/2),\ -0.778 - 0.627i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.034583579\)
\(L(\frac12)\) \(\approx\) \(1.034583579\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 2.59i)T \)
good5 \( 1 + (-1.22 - 2.12i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.67 + 2.12i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.73iT - 13T^{2} \)
17 \( 1 + (2.44 - 4.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.5 - 2.59i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.48iT - 29T^{2} \)
31 \( 1 + (-1.5 - 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 12.2T + 41T^{2} \)
43 \( 1 - 11T + 43T^{2} \)
47 \( 1 + (1.22 + 2.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.34 + 4.24i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.44 - 4.24i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 - 1.73i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.7iT - 71T^{2} \)
73 \( 1 + (-13.5 - 7.79i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 + (-7.34 - 12.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.46iT - 97T^{2} \)
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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

−10.35114819479896763805486386282, −9.579491681616927301989561511860, −8.498533504858456367193036977264, −8.016206767619228802205598331170, −6.55264493989360094446862009763, −6.21877742617943630113763350783, −5.30033420940546620338107280662, −3.99392036709611092284993164219, −2.74002355058970170247253193875, −2.10570633483515364680817839395, 0.43662848751911183638056613547, 1.93853191716278489484572768193, 3.20709162832593535650803844962, 4.74311949036853756496108450915, 4.91875057376468010794363206929, 6.24056060418528971947038944264, 7.20226163126354531743418851038, 7.909136529473175596026606068299, 8.989654849722098273917684273814, 9.527054934202684853032740921468

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