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Properties

Label 1-2675-2675.389-r1-0-0
Degree $1$
Conductor $2675$
Sign $-0.972 - 0.233i$
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  + (−0.835 − 0.548i)2-s + (−0.900 − 0.435i)3-s + (0.397 + 0.917i)4-s + (0.513 + 0.858i)6-s + (0.889 + 0.456i)7-s + (0.171 − 0.985i)8-s + (0.620 + 0.783i)9-s + (0.100 + 0.994i)11-s + (0.0414 − 0.999i)12-s + (0.275 − 0.961i)13-s + (−0.493 − 0.869i)14-s + (−0.683 + 0.729i)16-s + (−0.733 + 0.679i)17-s + (−0.0887 − 0.996i)18-s + (0.240 + 0.970i)19-s + ⋯
L(s)  = 1  + (−0.835 − 0.548i)2-s + (−0.900 − 0.435i)3-s + (0.397 + 0.917i)4-s + (0.513 + 0.858i)6-s + (0.889 + 0.456i)7-s + (0.171 − 0.985i)8-s + (0.620 + 0.783i)9-s + (0.100 + 0.994i)11-s + (0.0414 − 0.999i)12-s + (0.275 − 0.961i)13-s + (−0.493 − 0.869i)14-s + (−0.683 + 0.729i)16-s + (−0.733 + 0.679i)17-s + (−0.0887 − 0.996i)18-s + (0.240 + 0.970i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2675\)    =    \(5^{2} \cdot 107\)
Sign: $-0.972 - 0.233i$
Analytic conductor: \(287.468\)
Root analytic conductor: \(287.468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2675} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2675,\ (1:\ ),\ -0.972 - 0.233i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.009897736344 - 0.08346264314i\)
\(L(\frac12)\) \(\approx\) \(0.009897736344 - 0.08346264314i\)
\(L(1)\) \(\approx\) \(0.5527134384 - 0.07785899223i\)
\(L(1)\) \(\approx\) \(0.5527134384 - 0.07785899223i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
107 \( 1 \)
good2 \( 1 + (-0.835 - 0.548i)T \)
3 \( 1 + (-0.900 - 0.435i)T \)
7 \( 1 + (0.889 + 0.456i)T \)
11 \( 1 + (0.100 + 0.994i)T \)
13 \( 1 + (0.275 - 0.961i)T \)
17 \( 1 + (-0.733 + 0.679i)T \)
19 \( 1 + (0.240 + 0.970i)T \)
23 \( 1 + (-0.910 + 0.413i)T \)
29 \( 1 + (0.966 - 0.257i)T \)
31 \( 1 + (-0.286 + 0.958i)T \)
37 \( 1 + (0.386 - 0.922i)T \)
41 \( 1 + (0.801 - 0.597i)T \)
43 \( 1 + (0.0296 - 0.999i)T \)
47 \( 1 + (-0.815 + 0.578i)T \)
53 \( 1 + (-0.353 + 0.935i)T \)
59 \( 1 + (-0.543 + 0.839i)T \)
61 \( 1 + (-0.159 - 0.987i)T \)
67 \( 1 + (0.989 + 0.141i)T \)
71 \( 1 + (-0.692 - 0.721i)T \)
73 \( 1 + (-0.895 - 0.446i)T \)
79 \( 1 + (-0.0651 + 0.997i)T \)
83 \( 1 + (-0.00592 - 0.999i)T \)
89 \( 1 + (-0.974 - 0.223i)T \)
97 \( 1 + (-0.648 - 0.761i)T \)
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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

−19.30868752091359915435883531364, −18.38595482733684334720008030844, −17.97793007767205306663597048050, −17.37330872178401004261623079217, −16.49802194316192595637399872162, −16.260902824254809029384891275730, −15.49505535572778550554083237335, −14.62033616255902314092832824895, −13.98475097803334003178193919557, −13.21926242087223572453509099562, −11.72452462232337117034080311972, −11.3837109208786365618580239610, −10.96784062149706244309827872171, −10.02956907585450177458536716750, −9.36019699989151968605653370492, −8.585318674365109360129400771035, −7.868495854399824236888691819295, −6.80796346809204493712341847832, −6.46079817527091198318013546413, −5.53822718114280093450601892230, −4.7079660629226613388645090376, −4.2021512647167740802582421335, −2.741148646181741698567131277461, −1.551270715251932069042434789097, −0.787789168728074063611127878871, 0.02714360692577175804523025778, 1.24000415087198589585928588110, 1.76751384311201994583746130086, 2.55392990039773950791304617889, 3.874026960161811025659320495157, 4.62424993134347067110899750665, 5.62967910847559308006297894143, 6.32313468691464324850914317570, 7.354406393733568043703402768920, 7.85929413223557110054269987751, 8.5320204845888369092344299415, 9.522278540935371029814701839461, 10.449353032008119580596719597587, 10.73807599442532228903695675043, 11.631847392225551401909260573371, 12.378567880658339399743529577682, 12.538101415670727033660663276083, 13.60708622066288821393589078983, 14.60694456301424981143277689667, 15.6842116512866846595258024200, 15.94612898681689976129592736411, 17.20785702877491107896108063155, 17.49175229398030981076593762947, 18.08987309950683148135559905808, 18.48282142337903602810107821740

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