↑

Properties

Label 1-1275-1275.1103-r0-0-0
Degree $1$
Conductor $1275$
Sign $0.572 + 0.819i$
Analytic cond. $5.92107$
Root an. cond. $5.92107$
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  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (0.707 − 0.707i)7-s + (−0.309 + 0.951i)8-s + (−0.156 − 0.987i)11-s + (0.587 + 0.809i)13-s + (0.987 − 0.156i)14-s + (−0.809 + 0.587i)16-s + (−0.951 − 0.309i)19-s + (0.453 − 0.891i)22-s + (0.987 − 0.156i)23-s + i·26-s + (0.891 + 0.453i)28-s + (0.891 + 0.453i)29-s + (0.453 + 0.891i)31-s − 32-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (0.707 − 0.707i)7-s + (−0.309 + 0.951i)8-s + (−0.156 − 0.987i)11-s + (0.587 + 0.809i)13-s + (0.987 − 0.156i)14-s + (−0.809 + 0.587i)16-s + (−0.951 − 0.309i)19-s + (0.453 − 0.891i)22-s + (0.987 − 0.156i)23-s + i·26-s + (0.891 + 0.453i)28-s + (0.891 + 0.453i)29-s + (0.453 + 0.891i)31-s − 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1275\)    =    \(3 \cdot 5^{2} \cdot 17\)
Sign: $0.572 + 0.819i$
Analytic conductor: \(5.92107\)
Root analytic conductor: \(5.92107\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1275} (1103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1275,\ (0:\ ),\ 0.572 + 0.819i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.466255886 + 1.285161224i\)
\(L(\frac12)\) \(\approx\) \(2.466255886 + 1.285161224i\)
\(L(1)\) \(\approx\) \(1.708430391 + 0.6065788597i\)
\(L(1)\) \(\approx\) \(1.708430391 + 0.6065788597i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.809 + 0.587i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (-0.156 - 0.987i)T \)
13 \( 1 + (0.587 + 0.809i)T \)
19 \( 1 + (-0.951 - 0.309i)T \)
23 \( 1 + (0.987 - 0.156i)T \)
29 \( 1 + (0.891 + 0.453i)T \)
31 \( 1 + (0.453 + 0.891i)T \)
37 \( 1 + (-0.987 - 0.156i)T \)
41 \( 1 + (0.987 + 0.156i)T \)
43 \( 1 + T \)
47 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.587 + 0.809i)T \)
61 \( 1 + (-0.156 - 0.987i)T \)
67 \( 1 + (-0.951 - 0.309i)T \)
71 \( 1 + (0.891 + 0.453i)T \)
73 \( 1 + (-0.156 - 0.987i)T \)
79 \( 1 + (-0.453 + 0.891i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (0.453 - 0.891i)T \)
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

−20.84816390572808252310950328521, −20.562659554473887975684822800017, −19.38124455717360167795853574431, −18.87489068798070544770163567509, −17.88906179758885629595999504565, −17.35135374269552543872732558104, −15.945805744533586288971155071662, −15.26736824048650106415171010749, −14.84061627106108997330872535841, −13.94566692946916886464616122546, −12.999719855992993169252379958145, −12.464916884659846620695216755766, −11.6950284055673482713034576269, −10.85519302247168426629831805370, −10.234390818209435114833870319320, −9.23574127171177727903781610335, −8.335977639727422361611674401819, −7.32713311326305902274533559754, −6.25626133373158548587365274776, −5.51541544852910008727356639704, −4.71270631273308998741356790338, −3.957110036344872210596289846219, −2.74351251257180124157373871052, −2.11399574822600787905311089664, −1.0287419939211596996364206908, 1.11367099786305304416136834439, 2.41274067251599498757017219857, 3.434423184090237220081247699026, 4.2854920475755645544085683134, 4.95344887914540146167338656662, 5.978191876670385715911490341985, 6.75166029576787889100562905523, 7.48252990126782836558201124911, 8.539836215529606940687729768185, 8.887457442105854559111862152793, 10.67282389729030653276043757067, 10.97041644208826732876380654671, 11.93794657044448223303845259055, 12.79874480548468053184583197015, 13.79173417891527365402427157613, 13.99344695872606081355098286251, 14.90887378462565223509313788858, 15.80088001714937817072885168403, 16.466889799720783139092788593, 17.16751651154222330694045675232, 17.82602863152948979175178125058, 18.89076651332812585239314288187, 19.69752804179022280821650555156, 20.775273142771680492389941710931, 21.26087892751110821004250181478

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