↑

Properties

Label 1-1755-1755.1064-r0-0-0
Degree $1$
Conductor $1755$
Sign $0.627 - 0.778i$
Analytic cond. $8.15018$
Root an. cond. $8.15018$
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  + (0.342 − 0.939i)2-s + (−0.766 − 0.642i)4-s + (−0.984 + 0.173i)7-s + (−0.866 + 0.5i)8-s + (−0.984 + 0.173i)11-s + (−0.173 + 0.984i)14-s + (0.173 + 0.984i)16-s − 17-s + (−0.866 − 0.5i)19-s + (−0.173 + 0.984i)22-s + (−0.173 + 0.984i)23-s + (0.866 + 0.5i)28-s + (0.939 + 0.342i)29-s + (−0.342 − 0.939i)31-s + (0.984 + 0.173i)32-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)2-s + (−0.766 − 0.642i)4-s + (−0.984 + 0.173i)7-s + (−0.866 + 0.5i)8-s + (−0.984 + 0.173i)11-s + (−0.173 + 0.984i)14-s + (0.173 + 0.984i)16-s − 17-s + (−0.866 − 0.5i)19-s + (−0.173 + 0.984i)22-s + (−0.173 + 0.984i)23-s + (0.866 + 0.5i)28-s + (0.939 + 0.342i)29-s + (−0.342 − 0.939i)31-s + (0.984 + 0.173i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1755\)    =    \(3^{3} \cdot 5 \cdot 13\)
Sign: $0.627 - 0.778i$
Analytic conductor: \(8.15018\)
Root analytic conductor: \(8.15018\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1755} (1064, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1755,\ (0:\ ),\ 0.627 - 0.778i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8420285237 - 0.4027183045i\)
\(L(\frac12)\) \(\approx\) \(0.8420285237 - 0.4027183045i\)
\(L(1)\) \(\approx\) \(0.7635515584 - 0.3808115318i\)
\(L(1)\) \(\approx\) \(0.7635515584 - 0.3808115318i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.342 - 0.939i)T \)
7 \( 1 + (-0.984 + 0.173i)T \)
11 \( 1 + (-0.984 + 0.173i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (0.939 + 0.342i)T \)
31 \( 1 + (-0.342 - 0.939i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (0.642 - 0.766i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (-0.342 + 0.939i)T \)
53 \( 1 + T \)
59 \( 1 + (0.984 + 0.173i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (0.642 - 0.766i)T \)
71 \( 1 - iT \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (-0.984 + 0.173i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.342 + 0.939i)T \)
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

−20.366401692897276345106581548889, −19.58317404113521877339747903127, −18.63797315364035889610924954919, −18.14211817410369296234603734279, −17.239909943238348148780481295836, −16.43888006298673179218715496553, −16.00164360669087784020837614571, −15.27102359891742525989559874574, −14.50984430433695532701852618272, −13.63557695651261917403249533097, −12.945124847486101695749902709099, −12.60479304899887060578542949152, −11.44327675897560679253169241237, −10.32675073155430507324678150934, −9.79757340850628380391162657308, −8.58320610749617823757107621905, −8.29866676759319782490322383629, −7.10612475247065867684100894391, −6.533547132488336837248330095258, −5.86362042351991117869297261957, −4.8449338570551211456395952121, −4.14368138793980929278227608042, −3.1592362303437802327912507874, −2.361556362130699106124464954551, −0.48749088246798640646088699308, 0.62844994490457901983709949630, 2.1476471817801783666097845720, 2.60389967012090448019908815010, 3.61923633523759062750055052918, 4.40017028439812168896667492386, 5.32485601490552505482041831449, 6.10171635441988936649228903989, 6.97655705593735538705515286000, 8.14512040900776612684660244433, 9.04245267388146644278024056549, 9.65052645087653966494073076527, 10.47599560788494374502026172003, 11.07008959378380560586296184283, 11.96606855010541427768194709938, 12.84341726852185532548057139173, 13.18358777601939641822278105734, 13.88760516518563930747826233822, 15.060854093125542933054791226600, 15.48860094666526895696618826032, 16.37073793849912378470075939109, 17.48220686975746318843617982446, 18.10088722553168703542076185686, 18.82432443028046010571082218983, 19.69886565089371084278400419812, 19.88869796078026294630115227497

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