↑

Properties

Label 2-275-11.5-c1-0-0
Degree $2$
Conductor $275$
Sign $0.999 + 0.0147i$
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  + (−0.854 − 2.63i)2-s + (−1.32 − 0.964i)3-s + (−4.56 + 3.31i)4-s + (−1.40 + 4.31i)6-s + (−1.53 + 1.11i)7-s + (8.16 + 5.92i)8-s + (−0.0944 − 0.290i)9-s + (0.102 + 3.31i)11-s + 9.27·12-s + (−0.638 − 1.96i)13-s + (4.24 + 3.08i)14-s + (5.12 − 15.7i)16-s + (−1.12 + 3.44i)17-s + (−0.683 + 0.496i)18-s + (1.03 + 0.752i)19-s + ⋯
L(s)  = 1  + (−0.604 − 1.85i)2-s + (−0.766 − 0.557i)3-s + (−2.28 + 1.65i)4-s + (−0.572 + 1.76i)6-s + (−0.580 + 0.421i)7-s + (2.88 + 2.09i)8-s + (−0.0314 − 0.0968i)9-s + (0.0309 + 0.999i)11-s + 2.67·12-s + (−0.177 − 0.544i)13-s + (1.13 + 0.824i)14-s + (1.28 − 3.94i)16-s + (−0.271 + 0.836i)17-s + (−0.161 + 0.117i)18-s + (0.237 + 0.172i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0147i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.999 + 0.0147i$
Analytic conductor: \(2.19588\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ 0.999 + 0.0147i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.227987 - 0.00168525i\)
\(L(\frac12)\) \(\approx\) \(0.227987 - 0.00168525i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.102 - 3.31i)T \)
good2 \( 1 + (0.854 + 2.63i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (1.32 + 0.964i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (1.53 - 1.11i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.638 + 1.96i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.12 - 3.44i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.03 - 0.752i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 3.36T + 23T^{2} \)
29 \( 1 + (0.910 - 0.661i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.42 - 4.37i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.45 - 1.05i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (3.73 + 2.71i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 0.0110T + 43T^{2} \)
47 \( 1 + (-7.46 - 5.42i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.02 - 6.22i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (4.43 - 3.21i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.27 + 6.99i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 7.01T + 67T^{2} \)
71 \( 1 + (3.68 - 11.3i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (2.44 - 1.77i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.89 + 5.82i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (4.52 - 13.9i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 + (5.34 + 16.4i)T + (-78.4 + 57.0i)T^{2} \)
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

−12.04788473102027361685598588994, −10.98477711939197607198410170238, −10.18078785033740667100330442868, −9.395585616311843668433342052477, −8.406367721258308073538202004714, −7.18862777366954133791275546910, −5.66383712279829021405208962561, −4.24342951566121055573123294113, −2.93237206300191531550731389566, −1.53071475283838975807169107557, 0.24491148271050825005415600688, 4.10634832619318248088645979728, 5.17242056860080246112805053636, 6.00844315420659617695329553047, 6.85914287885108092906222089294, 7.87108256736025539589274747704, 8.919010169561413870405579958740, 9.785787298617188499314167554667, 10.53547401317754411951852695536, 11.66296230980716520361532229934

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