↑

Properties

Label 1-5239-5239.1249-r0-0-0
Degree $1$
Conductor $5239$
Sign $-0.675 - 0.737i$
Analytic cond. $24.3298$
Root an. cond. $24.3298$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.926 + 0.377i)2-s + (−0.293 + 0.955i)3-s + (0.715 + 0.698i)4-s + (−0.996 + 0.0804i)5-s + (−0.632 + 0.774i)6-s + (−0.00805 + 0.999i)7-s + (0.399 + 0.916i)8-s + (−0.827 − 0.561i)9-s + (−0.953 − 0.301i)10-s + (0.899 + 0.435i)11-s + (−0.877 + 0.478i)12-s + (−0.384 + 0.923i)14-s + (0.215 − 0.976i)15-s + (0.0241 + 0.999i)16-s + (0.594 − 0.804i)17-s + (−0.554 − 0.832i)18-s + ⋯
L(s)  = 1  + (0.926 + 0.377i)2-s + (−0.293 + 0.955i)3-s + (0.715 + 0.698i)4-s + (−0.996 + 0.0804i)5-s + (−0.632 + 0.774i)6-s + (−0.00805 + 0.999i)7-s + (0.399 + 0.916i)8-s + (−0.827 − 0.561i)9-s + (−0.953 − 0.301i)10-s + (0.899 + 0.435i)11-s + (−0.877 + 0.478i)12-s + (−0.384 + 0.923i)14-s + (0.215 − 0.976i)15-s + (0.0241 + 0.999i)16-s + (0.594 − 0.804i)17-s + (−0.554 − 0.832i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5239\)    =    \(13^{2} \cdot 31\)
Sign: $-0.675 - 0.737i$
Analytic conductor: \(24.3298\)
Root analytic conductor: \(24.3298\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5239} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5239,\ (0:\ ),\ -0.675 - 0.737i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.7491571015 + 1.701875959i\)
\(L(\frac12)\) \(\approx\) \(-0.7491571015 + 1.701875959i\)
\(L(1)\) \(\approx\) \(0.8932334670 + 1.051403998i\)
\(L(1)\) \(\approx\) \(0.8932334670 + 1.051403998i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.926 + 0.377i)T \)
3 \( 1 + (-0.293 + 0.955i)T \)
5 \( 1 + (-0.996 + 0.0804i)T \)
7 \( 1 + (-0.00805 + 0.999i)T \)
11 \( 1 + (0.899 + 0.435i)T \)
17 \( 1 + (0.594 - 0.804i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (0.926 + 0.377i)T \)
37 \( 1 + (-0.632 + 0.774i)T \)
41 \( 1 + (0.513 + 0.857i)T \)
43 \( 1 + (-0.892 - 0.450i)T \)
47 \( 1 + (-0.989 + 0.144i)T \)
53 \( 1 + (-0.993 - 0.112i)T \)
59 \( 1 + (0.184 + 0.982i)T \)
61 \( 1 + (-0.748 - 0.663i)T \)
67 \( 1 + (-0.845 + 0.534i)T \)
71 \( 1 + (-0.657 + 0.753i)T \)
73 \( 1 + (0.899 + 0.435i)T \)
79 \( 1 + (-0.962 - 0.270i)T \)
83 \( 1 + (-0.293 - 0.955i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.958 + 0.285i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.37155597831288434870677509806, −16.93325692170635709494423071361, −16.165493442071226204476234339223, −15.510974351475653764893710013256, −14.59259946489583104569323225471, −14.16841915118582844436501443475, −13.41857113160824874369293343559, −12.93006645441343593707523808417, −12.14681297135341556379451732060, −11.711463550762934231121019793119, −11.010711568125025290679946467435, −10.65671377051435892044027039764, −9.547394943055326146833302365180, −8.54492814754046606154590477404, −7.7524556071073013304181621916, −7.15602013548518341642916453916, −6.60433244652375612789778319428, −5.93372724675421148859598205451, −4.96279678167049813646150262553, −4.34522434457686619088876305977, −3.468348323549895731854547081294, −3.10020572835786868138767549722, −1.78504890343886814355500873468, −1.16983950268883742124605518731, −0.37745117271895060498838204685, 1.37318701460950286033371268724, 2.67940537988333779198533622704, 3.22061680585133882421042557840, 3.8661335245685086511315552020, 4.8320082179135224907786028813, 4.88540433034752308374106701038, 6.0597919084435335872969514749, 6.49649223523719808351582162066, 7.37266432736091520128254425019, 8.33394498407483168350931451559, 8.69385515126735680407306567427, 9.68026193873664661002064725957, 10.45823737591102352883820078320, 11.3199774291216766380648256160, 11.95578162927048386723820152706, 12.08002728463280706254251759798, 12.87403830510111425310156086045, 14.14843592848855654082569771997, 14.622990626428001854961613459400, 14.97587174219608140688364078290, 15.72992724981338025824314837465, 16.23098698620249353257397733369, 16.66470309262952798276256885175, 17.47086187085530085940712831425, 18.35384735704732495499701161497

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