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Properties

Label 1-2952-2952.2867-r1-0-0
Degree $1$
Conductor $2952$
Sign $-0.832 - 0.553i$
Analytic cond. $317.236$
Root an. cond. $317.236$
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  + (0.866 − 0.5i)5-s + (0.965 − 0.258i)7-s + (−0.965 + 0.258i)11-s + (−0.258 + 0.965i)13-s + (−0.707 + 0.707i)17-s + (0.707 − 0.707i)19-s + (0.5 + 0.866i)23-s + (0.5 − 0.866i)25-s + (−0.965 + 0.258i)29-s + (−0.5 − 0.866i)31-s + (0.707 − 0.707i)35-s − 37-s + (0.866 + 0.5i)43-s + (0.258 + 0.965i)47-s + (0.866 − 0.5i)49-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)5-s + (0.965 − 0.258i)7-s + (−0.965 + 0.258i)11-s + (−0.258 + 0.965i)13-s + (−0.707 + 0.707i)17-s + (0.707 − 0.707i)19-s + (0.5 + 0.866i)23-s + (0.5 − 0.866i)25-s + (−0.965 + 0.258i)29-s + (−0.5 − 0.866i)31-s + (0.707 − 0.707i)35-s − 37-s + (0.866 + 0.5i)43-s + (0.258 + 0.965i)47-s + (0.866 − 0.5i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2952\)    =    \(2^{3} \cdot 3^{2} \cdot 41\)
Sign: $-0.832 - 0.553i$
Analytic conductor: \(317.236\)
Root analytic conductor: \(317.236\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2952} (2867, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2952,\ (1:\ ),\ -0.832 - 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2612764942 - 0.8644589394i\)
\(L(\frac12)\) \(\approx\) \(0.2612764942 - 0.8644589394i\)
\(L(1)\) \(\approx\) \(1.123302639 - 0.1014706895i\)
\(L(1)\) \(\approx\) \(1.123302639 - 0.1014706895i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
41 \( 1 \)
good5 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (0.965 - 0.258i)T \)
11 \( 1 + (-0.965 + 0.258i)T \)
13 \( 1 + (-0.258 + 0.965i)T \)
17 \( 1 + (-0.707 + 0.707i)T \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.965 + 0.258i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (0.258 + 0.965i)T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.866 - 0.5i)T \)
67 \( 1 + (-0.965 - 0.258i)T \)
71 \( 1 + (0.707 + 0.707i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.965 + 0.258i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (-0.707 - 0.707i)T \)
97 \( 1 + (-0.258 - 0.965i)T \)
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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

−18.90467695270333937914133930979, −18.379998586133463390158051440784, −17.916812184104852111889010165360, −17.31351448693750503665687150765, −16.467470741691182456440798383387, −15.54361947527538558251150271265, −15.029085270236639136392212900308, −14.187246671282944445291983375594, −13.74231525140847510888682960871, −12.89016242306531889437119944627, −12.21963007031090772260486614547, −11.16574077179376181366784556056, −10.702077021619001037761903065129, −10.08812163422563281475306779724, −9.15254442555904522858711483704, −8.460356420450967797230611355996, −7.58511804563301717338479463160, −7.03958612510793409342296230230, −5.87903498491201777552082392815, −5.35333562446942488193270253812, −4.8136889873226542350078082382, −3.49301626681978919669409151859, −2.66028069935013713397491979115, −2.07928152881759282880560794572, −1.053434462061881948291862786939, 0.13032064470807099353609308047, 1.427457362630754900493145949339, 1.89797218811933435722220757875, 2.77666347907453148607563197833, 4.04937527993176253441656023292, 4.79461115162257702754358953120, 5.32978276065619613909244172198, 6.135440673222988842808312861383, 7.2108431114183771080977413316, 7.69218347152655254562224185908, 8.75211452971153786798188178745, 9.24338731480571720582482743242, 10.015304647158504514023185096352, 10.9445283606808709736514124457, 11.34432316893418965726652543995, 12.38723959903176537560360020557, 13.13714672498580881202363729251, 13.635131120203980242938427014917, 14.34953886797888107779181625821, 15.10717446851463296379090911631, 15.84722491341570854683711407193, 16.69663189098744254493931143897, 17.38192728391478411741032353911, 17.76899968418030161249109840326, 18.497771527102551033582685235554

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