Properties

Label 2-275-11.3-c1-0-2
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: $\chi_{275} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ -0.124 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.447799 + 0.507590i\)
\(L(\frac12)\) \(\approx\) \(0.447799 + 0.507590i\)
\(L(\frac{3}{2})\) 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

−12.31787943536636848214331123299, −10.63700736818109479091465258956, −9.983589390737898861110585751566, −9.300186051061108623022824653199, −8.093900311227876988466049201446, −7.57895800528886288928094665327, −6.37630337836990307845069665485, −5.57355343014766527293174055807, −3.73819948293842009161262154202, −1.45676877926644045864207070199, 0.922238857825897360608938233955, 2.51611835659110189073815001598, 3.86972161378854147595330468278, 5.57163064706683121309567497830, 7.24394477138721547648666540852, 7.936723437013820793445011966783, 8.947556182849227377531279601722, 9.967267971299977263758724613776, 10.44393528579238680745756156719, 11.46938661752783524339151168278

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