↑

Properties

Label 2-275-11.4-c1-0-8
Degree $2$
Conductor $275$
Sign $0.124 - 0.992i$
Analytic cond. $2.19588$
Root an. cond. $1.48185$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.85 + 1.34i)2-s + (−0.0131 + 0.0403i)3-s + (1.00 + 3.10i)4-s + (−0.0786 + 0.0571i)6-s + (0.352 + 1.08i)7-s + (−0.894 + 2.75i)8-s + (2.42 + 1.76i)9-s + (−1.53 − 2.94i)11-s − 0.138·12-s + (−2.27 − 1.65i)13-s + (−0.807 + 2.48i)14-s + (−0.0933 + 0.0677i)16-s + (−6.25 + 4.54i)17-s + (2.12 + 6.54i)18-s + (1.01 − 3.12i)19-s + ⋯
L(s)  = 1  + (1.31 + 0.953i)2-s + (−0.00756 + 0.0232i)3-s + (0.503 + 1.55i)4-s + (−0.0321 + 0.0233i)6-s + (0.133 + 0.409i)7-s + (−0.316 + 0.973i)8-s + (0.808 + 0.587i)9-s + (−0.461 − 0.887i)11-s − 0.0399·12-s + (−0.630 − 0.458i)13-s + (−0.215 + 0.664i)14-s + (−0.0233 + 0.0169i)16-s + (−1.51 + 1.10i)17-s + (0.500 + 1.54i)18-s + (0.232 − 0.716i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.124 - 0.992i$
Analytic conductor: \(2.19588\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ 0.124 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83943 + 1.62276i\)
\(L(\frac12)\) \(\approx\) \(1.83943 + 1.62276i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 + (1.53 + 2.94i)T \)
good2 \( 1 + (-1.85 - 1.34i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.0131 - 0.0403i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (-0.352 - 1.08i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (2.27 + 1.65i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (6.25 - 4.54i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.01 + 3.12i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 5.54T + 23T^{2} \)
29 \( 1 + (2.15 + 6.62i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.604 - 0.439i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.48 + 4.57i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.05 + 6.33i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 0.698T + 43T^{2} \)
47 \( 1 + (2.67 - 8.22i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (7.08 + 5.15i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.28 - 10.1i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (7.48 - 5.44i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 6.69T + 67T^{2} \)
71 \( 1 + (6.03 - 4.38i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.472 - 1.45i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-8.11 - 5.89i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.96 - 2.87i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 9.00T + 89T^{2} \)
97 \( 1 + (-4.79 - 3.48i)T + (29.9 + 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.61541440065188737753662094844, −11.33969635246998792048045579129, −10.44630335755928040164508043097, −8.963985905820228941523849062952, −7.86888261191454271084334056052, −7.01236024335791306061062200200, −5.94052540430765995083695298701, −5.03585622215402594020540744992, −4.11736586961360434702180877804, −2.61321796592868141768839537120, 1.75101054373141819522033451894, 3.12472383524083026532441364917, 4.47812846930301418427266269486, 4.93957345454736225745727415280, 6.59134126644160080354931313787, 7.41717436174178072956722585776, 9.200678908570544980359053359496, 10.09151072638226298803465893164, 10.99695856663029243579408346275, 11.85745037998371182613858865707

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