↑

Properties

Label 1-6025-6025.1038-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.824 + 0.566i$
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  + (0.544 + 0.838i)2-s + (0.838 + 0.544i)3-s + (−0.406 + 0.913i)4-s + i·6-s + (0.566 − 0.824i)7-s + (−0.987 + 0.156i)8-s + (0.406 + 0.913i)9-s + (−0.824 − 0.566i)11-s + (−0.838 + 0.544i)12-s + (−0.942 − 0.333i)13-s + (0.999 + 0.0261i)14-s + (−0.669 − 0.743i)16-s + (0.996 − 0.0784i)17-s + (−0.544 + 0.838i)18-s + (0.0261 + 0.999i)19-s + ⋯
L(s)  = 1  + (0.544 + 0.838i)2-s + (0.838 + 0.544i)3-s + (−0.406 + 0.913i)4-s + i·6-s + (0.566 − 0.824i)7-s + (−0.987 + 0.156i)8-s + (0.406 + 0.913i)9-s + (−0.824 − 0.566i)11-s + (−0.838 + 0.544i)12-s + (−0.942 − 0.333i)13-s + (0.999 + 0.0261i)14-s + (−0.669 − 0.743i)16-s + (0.996 − 0.0784i)17-s + (−0.544 + 0.838i)18-s + (0.0261 + 0.999i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.824 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.824 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.824 + 0.566i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (1038, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.824 + 0.566i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8372193188 + 2.696143184i\)
\(L(\frac12)\) \(\approx\) \(0.8372193188 + 2.696143184i\)
\(L(1)\) \(\approx\) \(1.303638313 + 1.064947737i\)
\(L(1)\) \(\approx\) \(1.303638313 + 1.064947737i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.544 + 0.838i)T \)
3 \( 1 + (0.838 + 0.544i)T \)
7 \( 1 + (0.566 - 0.824i)T \)
11 \( 1 + (-0.824 - 0.566i)T \)
13 \( 1 + (-0.942 - 0.333i)T \)
17 \( 1 + (0.996 - 0.0784i)T \)
19 \( 1 + (0.0261 + 0.999i)T \)
23 \( 1 + (0.852 - 0.522i)T \)
29 \( 1 + (-0.544 - 0.838i)T \)
31 \( 1 + (0.902 + 0.430i)T \)
37 \( 1 + (-0.958 - 0.284i)T \)
41 \( 1 + (-0.707 + 0.707i)T \)
43 \( 1 + (0.0784 + 0.996i)T \)
47 \( 1 + (0.891 + 0.453i)T \)
53 \( 1 + (-0.998 + 0.0523i)T \)
59 \( 1 + (-0.777 - 0.629i)T \)
61 \( 1 + (0.156 + 0.987i)T \)
67 \( 1 + (-0.0523 + 0.998i)T \)
71 \( 1 + (0.983 - 0.182i)T \)
73 \( 1 + (0.760 + 0.649i)T \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 + (0.978 + 0.207i)T \)
89 \( 1 + (0.983 - 0.182i)T \)
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

−17.65864707487262012580752179944, −17.07190913902189638755989312677, −15.58673046120775285822899445022, −15.26764823426426231929225957283, −14.80616272153993060271457994360, −13.95015797585152952210450561370, −13.62150524076845663516666730692, −12.59330819012384504350715544251, −12.39005167475728542597887458155, −11.759050587610413571902377033362, −10.91504402846113119264130731429, −10.13219701739662549715256068932, −9.40863562567395674208681125206, −8.96685846637179314361843512909, −8.14470229736151934915150352050, −7.357604478081802907368772756588, −6.75425542293311485105347516190, −5.629248136516721902881023109732, −5.06934475238516993305948372280, −4.49643451228278231538972997193, −3.312238757472090895546540233643, −2.9320144138747512686141794520, −2.02255737571172238551572259798, −1.76387177435714547591775720379, −0.536049528270065166227617313941, 0.99815335470498513369599245432, 2.28383395711481763144349481655, 3.04800676798129702619507557299, 3.58824949119222497695552754978, 4.425177678494102306383783621893, 5.01994123540838050644145470809, 5.532644928570450637996839244224, 6.58758100211770863057164775265, 7.48403777373706009929057454440, 7.91846768549361942420609285716, 8.248341914889485272418698506531, 9.2207993414359956166815055501, 10.00791886588237193827931414646, 10.49451709401339891315572026766, 11.382193596110260000670839672386, 12.31859247719581539817926327464, 12.95513051231662358010727619504, 13.64370489504155546444809364646, 14.21960019830817561401431746664, 14.601282782845108098473701060386, 15.26805817811659457095407309980, 15.91587904180119359079241342520, 16.63082785600219182195643865071, 17.0027446104339664088567395435, 17.74479716762767114745297821615

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