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Properties

Label 1-3192-3192.563-r0-0-0
Degree $1$
Conductor $3192$
Sign $0.0288 + 0.999i$
Analytic cond. $14.8235$
Root an. cond. $14.8235$
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)5-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s + 17-s + 23-s + (−0.5 + 0.866i)25-s + (0.5 + 0.866i)29-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)37-s + (0.5 − 0.866i)41-s + (−0.5 + 0.866i)43-s − 47-s + (0.5 − 0.866i)53-s + (−0.5 + 0.866i)55-s − 59-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)5-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s + 17-s + 23-s + (−0.5 + 0.866i)25-s + (0.5 + 0.866i)29-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)37-s + (0.5 − 0.866i)41-s + (−0.5 + 0.866i)43-s − 47-s + (0.5 − 0.866i)53-s + (−0.5 + 0.866i)55-s − 59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0288 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0288 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3192\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.0288 + 0.999i$
Analytic conductor: \(14.8235\)
Root analytic conductor: \(14.8235\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3192} (563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3192,\ (0:\ ),\ 0.0288 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.526822667 + 1.571532071i\)
\(L(\frac12)\) \(\approx\) \(1.526822667 + 1.571532071i\)
\(L(1)\) \(\approx\) \(1.241813134 + 0.4471853487i\)
\(L(1)\) \(\approx\) \(1.241813134 + 0.4471853487i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good5 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + T \)
23 \( 1 + T \)
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 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 - T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 - T \)
61 \( 1 + T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + T \)
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

−18.828996728318536660888371240325, −17.997132158634205858892593258057, −17.21043323492346427465026468635, −16.80509065723382990392510014694, −16.065052918470029142025406820847, −15.385030494449896363964855361209, −14.48733367526468464171731307803, −13.7803294113339383787510023560, −13.17854107160826852480069260162, −12.56014685824004920359249665280, −11.7488499836069496651264607440, −11.06305629289714055277124815953, −10.13117918165474183092463832453, −9.583614088505039181008854428, −8.65066209562603756431123009737, −8.279745287928362984613204833849, −7.368763550862842092426957421989, −6.249235154962535015173510617353, −5.75121584299082183597668260746, −5.07701323138485085760216538786, −4.11793845782541257555751420579, −3.31205369160048431069906680512, −2.444852758250923331034216107684, −1.23539428717525119825051702922, −0.74537437472203384643252373066, 1.2895081672691595000672806419, 1.86521111330208911414007078224, 2.98833654976795937350796044201, 3.52687169814654656495593138769, 4.615654799693267846441800147370, 5.30642057418047418352140092214, 6.40302787710754958904039385080, 6.764108407728408642963942647446, 7.48076811370854159044614526039, 8.50773191552076049732098086532, 9.28627181380539157602260438935, 9.946033491125017781087512735950, 10.57345280019030913470835956692, 11.37446249871775066239372123340, 12.03919551078433094416834631880, 12.81611459560399370887054852775, 13.68787938330022302224783133633, 14.32563448785847193397607726230, 14.76364934548583539439180637832, 15.56174312804054428147021971962, 16.4334455305118362965433625427, 17.08122352021114159370344818424, 17.81743792455859835500919756841, 18.35194279169889063884849833289, 19.14496155021566410076064093340

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