↑

Properties

Label 1-605-605.263-r0-0-0
Degree $1$
Conductor $605$
Sign $0.824 + 0.566i$
Analytic cond. $2.80960$
Root an. cond. $2.80960$
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  + (−0.755 − 0.654i)2-s + i·3-s + (0.142 + 0.989i)4-s + (0.654 − 0.755i)6-s + (0.281 − 0.959i)7-s + (0.540 − 0.841i)8-s − 9-s + (−0.989 + 0.142i)12-s + (−0.989 + 0.142i)13-s + (−0.841 + 0.540i)14-s + (−0.959 + 0.281i)16-s + (0.909 + 0.415i)17-s + (0.755 + 0.654i)18-s + (0.415 + 0.909i)19-s + (0.959 + 0.281i)21-s + ⋯
L(s)  = 1  + (−0.755 − 0.654i)2-s + i·3-s + (0.142 + 0.989i)4-s + (0.654 − 0.755i)6-s + (0.281 − 0.959i)7-s + (0.540 − 0.841i)8-s − 9-s + (−0.989 + 0.142i)12-s + (−0.989 + 0.142i)13-s + (−0.841 + 0.540i)14-s + (−0.959 + 0.281i)16-s + (0.909 + 0.415i)17-s + (0.755 + 0.654i)18-s + (0.415 + 0.909i)19-s + (0.959 + 0.281i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.824 + 0.566i$
Analytic conductor: \(2.80960\)
Root analytic conductor: \(2.80960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (0:\ ),\ 0.824 + 0.566i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8279689104 + 0.2569396610i\)
\(L(\frac12)\) \(\approx\) \(0.8279689104 + 0.2569396610i\)
\(L(1)\) \(\approx\) \(0.7468789182 + 0.05006248042i\)
\(L(1)\) \(\approx\) \(0.7468789182 + 0.05006248042i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.755 - 0.654i)T \)
3 \( 1 + iT \)
7 \( 1 + (0.281 - 0.959i)T \)
13 \( 1 + (-0.989 + 0.142i)T \)
17 \( 1 + (0.909 + 0.415i)T \)
19 \( 1 + (0.415 + 0.909i)T \)
23 \( 1 + (-0.281 - 0.959i)T \)
29 \( 1 + (0.415 + 0.909i)T \)
31 \( 1 + (-0.142 + 0.989i)T \)
37 \( 1 + (0.989 + 0.142i)T \)
41 \( 1 + (0.654 - 0.755i)T \)
43 \( 1 + (0.540 - 0.841i)T \)
47 \( 1 + (-0.755 + 0.654i)T \)
53 \( 1 + (0.281 - 0.959i)T \)
59 \( 1 + (0.654 + 0.755i)T \)
61 \( 1 + (0.654 + 0.755i)T \)
67 \( 1 + (0.755 + 0.654i)T \)
71 \( 1 + (0.415 + 0.909i)T \)
73 \( 1 + (0.281 + 0.959i)T \)
79 \( 1 + (0.841 - 0.540i)T \)
83 \( 1 + (-0.281 + 0.959i)T \)
89 \( 1 + (-0.415 + 0.909i)T \)
97 \( 1 + (0.540 - 0.841i)T \)
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

−23.27445897411806587368094386090, −22.41916086950846420659386527046, −21.33116013314776414324192178028, −20.07835952804572501809048030406, −19.44967409336344520999118616846, −18.70849023549542568135391906732, −17.987870149829965581708061569172, −17.4077874707621803936506228253, −16.50013050549646934945824581679, −15.43487416941270524828427011718, −14.70783971661433475465767765250, −13.92220652246272811522652566475, −12.87524895566363100425883691793, −11.77728051750863264038006291187, −11.29861675851658599868091241421, −9.73779357616412967025687426085, −9.21478130967390541757210339356, −7.900156047305176716672403604227, −7.70274922546549459806920812655, −6.5010525148962787484145324700, −5.68365501943133129920462333025, −4.91441754887354481955615903689, −2.80685643138624025262224138218, −2.008432999402881844799763955900, −0.7134224142580551492741387929, 1.014986780514776489060641835823, 2.45426427917532780364244674931, 3.55300849161957661137627457372, 4.26117271441132287602854789756, 5.334880897138237514357619142447, 6.85860765904695902220687948907, 7.84725994170941173073234620782, 8.62117864917672237579328737050, 9.785370709017141723738569953384, 10.21344741462194796404949656852, 10.9358686389106330686013359894, 11.93426749687006205797596827324, 12.71019246671082743220404300184, 14.15880630243829464260789832524, 14.59790955574982425613781782785, 16.08412690734728591235938071822, 16.55845765387931961390772568682, 17.26488114727365879027221894046, 18.08082081080274537605556313785, 19.26685575489797987292599220354, 19.9285571147038347797621186557, 20.68204528633554866327658131359, 21.24177708952049998465513518603, 22.14721724597630445947548221184, 22.8377252015017986096011812452

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