Properties

Label 1-2675-2675.644-r1-0-0
Degree $1$
Conductor $2675$
Sign $-0.851 - 0.524i$
Analytic cond. $287.468$
Root an. cond. $287.468$
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  + (0.842 − 0.538i)2-s + (0.969 − 0.246i)3-s + (0.419 − 0.907i)4-s + (0.683 − 0.729i)6-s + (0.829 − 0.558i)7-s + (−0.135 − 0.990i)8-s + (0.878 − 0.477i)9-s + (−0.780 − 0.625i)11-s + (0.182 − 0.983i)12-s + (0.112 + 0.993i)13-s + (0.397 − 0.917i)14-s + (−0.648 − 0.761i)16-s + (0.986 − 0.165i)17-s + (0.482 − 0.875i)18-s + (0.919 − 0.392i)19-s + ⋯
L(s)  = 1  + (0.842 − 0.538i)2-s + (0.969 − 0.246i)3-s + (0.419 − 0.907i)4-s + (0.683 − 0.729i)6-s + (0.829 − 0.558i)7-s + (−0.135 − 0.990i)8-s + (0.878 − 0.477i)9-s + (−0.780 − 0.625i)11-s + (0.182 − 0.983i)12-s + (0.112 + 0.993i)13-s + (0.397 − 0.917i)14-s + (−0.648 − 0.761i)16-s + (0.986 − 0.165i)17-s + (0.482 − 0.875i)18-s + (0.919 − 0.392i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2675\)    =    \(5^{2} \cdot 107\)
Sign: $-0.851 - 0.524i$
Analytic conductor: \(287.468\)
Root analytic conductor: \(287.468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2675} (644, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2675,\ (1:\ ),\ -0.851 - 0.524i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.870303472 - 6.595094259i\)
\(L(\frac12)\) \(\approx\) \(1.870303472 - 6.595094259i\)
\(L(1)\) \(\approx\) \(2.092691155 - 1.648925019i\)
\(L(1)\) \(\approx\) \(2.092691155 - 1.648925019i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
107 \( 1 \)
good2 \( 1 + (0.842 - 0.538i)T \)
3 \( 1 + (0.969 - 0.246i)T \)
7 \( 1 + (0.829 - 0.558i)T \)
11 \( 1 + (-0.780 - 0.625i)T \)
13 \( 1 + (0.112 + 0.993i)T \)
17 \( 1 + (0.986 - 0.165i)T \)
19 \( 1 + (0.919 - 0.392i)T \)
23 \( 1 + (-0.995 + 0.0946i)T \)
29 \( 1 + (0.801 + 0.597i)T \)
31 \( 1 + (-0.959 + 0.280i)T \)
37 \( 1 + (-0.966 + 0.257i)T \)
41 \( 1 + (0.989 - 0.141i)T \)
43 \( 1 + (0.937 + 0.348i)T \)
47 \( 1 + (0.884 - 0.467i)T \)
53 \( 1 + (0.252 - 0.967i)T \)
59 \( 1 + (0.297 - 0.954i)T \)
61 \( 1 + (-0.999 + 0.0355i)T \)
67 \( 1 + (-0.472 - 0.881i)T \)
71 \( 1 + (-0.639 - 0.768i)T \)
73 \( 1 + (-0.408 - 0.912i)T \)
79 \( 1 + (-0.159 - 0.987i)T \)
83 \( 1 + (0.765 - 0.643i)T \)
89 \( 1 + (0.124 + 0.992i)T \)
97 \( 1 + (-0.0177 + 0.999i)T \)
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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

−19.68392536029629949741893222705, −18.54692613223057279407891316929, −18.03337371215778589432496851383, −17.29361599496079233667417708879, −16.18695412374443140155199824502, −15.631747672481756948520259650242, −15.21551625707635371756992908217, −14.2687791153698897313953680995, −14.14240892290638715346882244503, −13.090307005811412127149800835391, −12.44567666266266436089948936854, −11.89626264810693037366414157907, −10.712940755399774684310566489, −10.1169452802289459632735300993, −9.06378218897004184679079718306, −8.26551474014755968822682472639, −7.62574238061728968763766421527, −7.40608110727707274591897565959, −5.76370597600971676243484792681, −5.50859498739323390377016435711, −4.5124568456055110760689104771, −3.850153122038938890967910798231, −2.85011678071917366361885810206, −2.38650867375132777321591398356, −1.32515364748937348364048690601, 0.62916098857871507370682136910, 1.48222432775883053736194991236, 2.16270856958478136701089461701, 3.15185540671551662443971501903, 3.69483952737749359570541824776, 4.59163683502324050552059994349, 5.286141398738924976974160945921, 6.26068722753156032188955452014, 7.28851358468453291975652851855, 7.69879598609101887028147483737, 8.73627647650907523550067663284, 9.49011394760132166769881781437, 10.36052535484859998046918956229, 10.92762826130185947524764416445, 11.89827486663505884190844703066, 12.36735543829128769562543933255, 13.47774163145916879619255923862, 13.8302785718706510019092399232, 14.3140013372603769957669170567, 14.94046802574326287530607243041, 16.10041109113231195859298012240, 16.19140316647803066653976201121, 17.75303095594360933807579178503, 18.35571736846937833121311109220, 19.03866783677810894506628166906

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