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Properties

Label 1-1005-1005.104-r1-0-0
Degree $1$
Conductor $1005$
Sign $-0.809 - 0.586i$
Analytic cond. $108.002$
Root an. cond. $108.002$
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  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)7-s + 8-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s − 14-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s − 22-s + (−0.5 + 0.866i)23-s + (0.5 + 0.866i)26-s + (0.5 − 0.866i)28-s + (0.5 + 0.866i)29-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)7-s + 8-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s − 14-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s − 22-s + (−0.5 + 0.866i)23-s + (0.5 + 0.866i)26-s + (0.5 − 0.866i)28-s + (0.5 + 0.866i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1005 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.809 - 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1005 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.809 - 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1005\)    =    \(3 \cdot 5 \cdot 67\)
Sign: $-0.809 - 0.586i$
Analytic conductor: \(108.002\)
Root analytic conductor: \(108.002\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1005} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1005,\ (1:\ ),\ -0.809 - 0.586i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2639307870 + 0.8140382401i\)
\(L(\frac12)\) \(\approx\) \(-0.2639307870 + 0.8140382401i\)
\(L(1)\) \(\approx\) \(0.6419646282 + 0.4651811874i\)
\(L(1)\) \(\approx\) \(0.6419646282 + 0.4651811874i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
67 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + T \)
59 \( 1 - T \)
61 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 - T \)
97 \( 1 + (0.5 - 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.98544910196766160351221012540, −20.0610562682498192807778414819, −19.59602303123059731898789486613, −18.62952876985764202590200517041, −18.02965813940786964661712586597, −17.07521877573717025457714068802, −16.59598556082062090866879744386, −15.690677151785407855360742381952, −14.17990950333597231262746751515, −13.85897142808846541563243755022, −13.02547594890409691510568632792, −11.8843680389405163291081490459, −11.25612016973508533943262358148, −10.72566422500767221209919906885, −9.71474978056121116861163560378, −8.83029909154230386801051276950, −8.25889839258445473253294013527, −7.14939273641520831560287150085, −6.389713779439913207107472785631, −4.76399657792741385861614364584, −4.21260704993458552770058250386, −3.22242327125988062301884995571, −2.14042345267422327810927215245, −1.102624330797520680842518000307, −0.23514288660044380612377333139, 1.350426844776301745748997540, 2.099256478720178743719186207925, 3.73798520358463958536137956400, 4.698316601293610804559560205424, 5.7055356861826991249998999349, 6.21794792973436975114753956324, 7.35970202848915083203144610824, 8.16269031082091989526776251363, 8.78065602989312987763541292802, 9.69095465024790703207159399535, 10.49151305736232160283025597680, 11.413553999726313290399887675165, 12.48637233101781009866461836922, 13.23471160964724766745870795938, 14.39949188324436025283068589735, 15.003396054732841569198983227160, 15.46170211207025807820858774160, 16.47191941325310329172507516142, 17.26644027095066766763170130485, 18.109302134808815231105407294066, 18.33610485128470500547476595495, 19.61810211839112373493346269350, 20.022529414486203418098995025272, 21.23991330855948460804163690634, 22.02022519168583213010024495044

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