↑

Properties

Label 1-275-275.4-r0-0-0
Degree $1$
Conductor $275$
Sign $-0.555 + 0.831i$
Analytic cond. $1.27709$
Root an. cond. $1.27709$
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  + (−0.309 + 0.951i)2-s + (−0.309 + 0.951i)3-s + (−0.809 − 0.587i)4-s + (−0.809 − 0.587i)6-s + (0.809 − 0.587i)7-s + (0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)12-s + (0.809 + 0.587i)13-s + (0.309 + 0.951i)14-s + (0.309 + 0.951i)16-s + (0.809 + 0.587i)17-s + (0.809 − 0.587i)18-s + (−0.809 + 0.587i)19-s + (0.309 + 0.951i)21-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.309 + 0.951i)3-s + (−0.809 − 0.587i)4-s + (−0.809 − 0.587i)6-s + (0.809 − 0.587i)7-s + (0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)12-s + (0.809 + 0.587i)13-s + (0.309 + 0.951i)14-s + (0.309 + 0.951i)16-s + (0.809 + 0.587i)17-s + (0.809 − 0.587i)18-s + (−0.809 + 0.587i)19-s + (0.309 + 0.951i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.555 + 0.831i$
Analytic conductor: \(1.27709\)
Root analytic conductor: \(1.27709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (0:\ ),\ -0.555 + 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4432168291 + 0.8294118050i\)
\(L(\frac12)\) \(\approx\) \(0.4432168291 + 0.8294118050i\)
\(L(1)\) \(\approx\) \(0.6518006296 + 0.5612530060i\)
\(L(1)\) \(\approx\) \(0.6518006296 + 0.5612530060i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (0.809 - 0.587i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (-0.309 + 0.951i)T \)
71 \( 1 + T \)
73 \( 1 - T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.809 - 0.587i)T \)
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

−25.32274819892369889662698556903, −24.53766636207577018387726832231, −23.29539781852094832112589521651, −22.76977631161275354773767817772, −21.54392532718293838458831313229, −20.79959060222927383383646164994, −19.803075047314283392697438930603, −18.78798517200948987103898106818, −18.262187159851789449814126819627, −17.4841067706282526353013172744, −16.54208024616025958114334570014, −14.93383645368621631263421990670, −13.85195526397235807763904819767, −12.95041357829399920708491927385, −12.15100374713238903658538173715, −11.27523697204468221327965553720, −10.57265067171900444669348096972, −8.97469030058600803168058411465, −8.28564692286798291576934416861, −7.27466644868812565502955285737, −5.75849710079221802511678226467, −4.76243607218084553047107303981, −3.100845198029956960543995526636, −2.0540876263115066354274144448, −0.9021090880485324616191337559, 1.31324716559644572164263950377, 3.71991257937499493266251765893, 4.52631646709517681061930665296, 5.58405835791077797814783126461, 6.53983069172199489110042973471, 7.90720918752892949602378217534, 8.701828153384202954458756783358, 9.83423443361516684217876206517, 10.65104947912418330715232285148, 11.59666328645571239178300791134, 13.24459331278488307018230785111, 14.321821738073958057220019888504, 14.9464024255117560820240025382, 15.90059063869917808620000093928, 16.93265084377516519543296695465, 17.23318874260774227240206286703, 18.45464604739938862687481221302, 19.45372942241366367451112863380, 20.86052620590664852527395240324, 21.334932976526643693232329495692, 22.695836062116559930147267304352, 23.34425608309974506991512693437, 24.03759181307202649403193033590, 25.296488765100089910456720476868, 26.108153411040655258382162880537

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