↑

Properties

Label 1-6025-6025.1498-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.997 + 0.0746i$
Analytic cond. $27.9799$
Root an. cond. $27.9799$
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  + (−0.838 − 0.544i)2-s + (−0.544 − 0.838i)3-s + (0.406 + 0.913i)4-s + i·6-s + (−0.983 − 0.182i)7-s + (0.156 − 0.987i)8-s + (−0.406 + 0.913i)9-s + (−0.182 + 0.983i)11-s + (0.544 − 0.838i)12-s + (0.902 + 0.430i)13-s + (0.725 + 0.688i)14-s + (−0.669 + 0.743i)16-s + (−0.649 − 0.760i)17-s + (0.838 − 0.544i)18-s + (0.688 + 0.725i)19-s + ⋯
L(s)  = 1  + (−0.838 − 0.544i)2-s + (−0.544 − 0.838i)3-s + (0.406 + 0.913i)4-s + i·6-s + (−0.983 − 0.182i)7-s + (0.156 − 0.987i)8-s + (−0.406 + 0.913i)9-s + (−0.182 + 0.983i)11-s + (0.544 − 0.838i)12-s + (0.902 + 0.430i)13-s + (0.725 + 0.688i)14-s + (−0.669 + 0.743i)16-s + (−0.649 − 0.760i)17-s + (0.838 − 0.544i)18-s + (0.688 + 0.725i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.997 + 0.0746i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (1498, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.997 + 0.0746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6876395370 + 0.02570961630i\)
\(L(\frac12)\) \(\approx\) \(0.6876395370 + 0.02570961630i\)
\(L(1)\) \(\approx\) \(0.5362476589 - 0.1617714939i\)
\(L(1)\) \(\approx\) \(0.5362476589 - 0.1617714939i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.838 - 0.544i)T \)
3 \( 1 + (-0.544 - 0.838i)T \)
7 \( 1 + (-0.983 - 0.182i)T \)
11 \( 1 + (-0.182 + 0.983i)T \)
13 \( 1 + (0.902 + 0.430i)T \)
17 \( 1 + (-0.649 - 0.760i)T \)
19 \( 1 + (0.688 + 0.725i)T \)
23 \( 1 + (-0.972 + 0.233i)T \)
29 \( 1 + (0.838 + 0.544i)T \)
31 \( 1 + (0.942 + 0.333i)T \)
37 \( 1 + (0.477 - 0.878i)T \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + (0.760 - 0.649i)T \)
47 \( 1 + (0.453 + 0.891i)T \)
53 \( 1 + (-0.0523 + 0.998i)T \)
59 \( 1 + (-0.629 - 0.777i)T \)
61 \( 1 + (-0.987 - 0.156i)T \)
67 \( 1 + (-0.998 + 0.0523i)T \)
71 \( 1 + (0.566 + 0.824i)T \)
73 \( 1 + (-0.0784 + 0.996i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + (0.978 - 0.207i)T \)
89 \( 1 + (0.566 + 0.824i)T \)
97 \( 1 + (-0.669 - 0.743i)T \)
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

−17.68283583125668833894191762404, −16.94049375616824245350870622199, −16.19538476876632738365931814647, −16.007145268521009854626603165166, −15.37673957965070621017684122687, −14.842801105252216131813213631833, −13.653189203301515127493720706214, −13.430320925476798001222371499443, −12.151871987941756354547564523922, −11.58283415210057752993998023820, −10.72967744289116586863653960829, −10.483630334326042490504729103341, −9.61979704220236501412855854628, −9.15368881787515748976386502063, −8.403894174582333709257676660460, −7.87972466981958108113671096828, −6.591361884488108429855176012643, −6.193417859749131374512329889956, −5.86694046208067883172544437051, −4.90060404255666622212762481729, −4.083041287755077212586178725, −3.16184335944615464471734175556, −2.53676462071599625830940059583, −1.08783511070630019158332025162, −0.41459241198756765460641895680, 0.71351165078487294620522551574, 1.458024136723193698136751099234, 2.29857171583900148984869830747, 2.91141690225524506972075080934, 3.90166202108722467888048108560, 4.61650229806909965732604171960, 5.86112993478780409106753020680, 6.36403216488994404427403531161, 7.16329048632519908388438413756, 7.50834878828235147681684196403, 8.35687153552823428646792918781, 9.201640606723392218859118946250, 9.71000323839301696466596045023, 10.57352273498373930798251384047, 10.95875232286430725995762740474, 11.99051530300480655139091935500, 12.250662142224395718813725027051, 12.829194129918671432636329309741, 13.765275818741571862232412729353, 13.96660120090024955431412556372, 15.56811105160307101440975842584, 16.02321539531441900459759328431, 16.40283201329335064887281318430, 17.375283675816523190427125465, 17.79337670286488557775493352700

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