↑

Properties

Label 2-55e2-275.202-c0-0-0
Degree $2$
Conductor $3025$
Sign $0.0561 + 0.998i$
Analytic cond. $1.50967$
Root an. cond. $1.22868$
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  + (1.95 − 0.309i)3-s − i·4-s + (−0.587 − 0.809i)5-s + (2.76 − 0.896i)9-s + (−0.309 − 1.95i)12-s + (−1.39 − 1.39i)15-s − 16-s + (−0.809 + 0.587i)20-s + (−0.142 + 0.896i)23-s + (−0.309 + 0.951i)25-s + (3.34 − 1.70i)27-s + (−0.363 + 1.11i)31-s + (−0.896 − 2.76i)36-s + (−1.26 + 0.642i)37-s + (−2.34 − 1.70i)45-s + ⋯
L(s)  = 1  + (1.95 − 0.309i)3-s − i·4-s + (−0.587 − 0.809i)5-s + (2.76 − 0.896i)9-s + (−0.309 − 1.95i)12-s + (−1.39 − 1.39i)15-s − 16-s + (−0.809 + 0.587i)20-s + (−0.142 + 0.896i)23-s + (−0.309 + 0.951i)25-s + (3.34 − 1.70i)27-s + (−0.363 + 1.11i)31-s + (−0.896 − 2.76i)36-s + (−1.26 + 0.642i)37-s + (−2.34 − 1.70i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0561 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0561 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $0.0561 + 0.998i$
Analytic conductor: \(1.50967\)
Root analytic conductor: \(1.22868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :0),\ 0.0561 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.246124815\)
\(L(\frac12)\) \(\approx\) \(2.246124815\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.587 + 0.809i)T \)
11 \( 1 \)
good2 \( 1 + iT^{2} \)
3 \( 1 + (-1.95 + 0.309i)T + (0.951 - 0.309i)T^{2} \)
7 \( 1 + (-0.587 - 0.809i)T^{2} \)
13 \( 1 + (0.587 + 0.809i)T^{2} \)
17 \( 1 + (-0.951 + 0.309i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.142 - 0.896i)T + (-0.951 - 0.309i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (1.26 - 0.642i)T + (0.587 - 0.809i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.221 + 1.39i)T + (-0.951 + 0.309i)T^{2} \)
53 \( 1 + (0.0489 - 0.309i)T + (-0.951 - 0.309i)T^{2} \)
59 \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (0.809 - 1.58i)T + (-0.587 - 0.809i)T^{2} \)
71 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.951 - 0.309i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.951 - 0.309i)T^{2} \)
89 \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.309 + 0.0489i)T + (0.951 + 0.309i)T^{2} \)
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

−8.815632079453234974099260267378, −8.159367276445081788248801208261, −7.31932150440379682269791317162, −6.82041544767806376710667673396, −5.54710221177656586826165125611, −4.67203428780980678386831699952, −3.88278037776700721803782173701, −3.08304695536849961939732195997, −1.91340615839899174067193735705, −1.24019861644498783536674038247, 2.11781877885894961313147104463, 2.69740364817472210497232145274, 3.54019106839205913222467763455, 3.94431172594984307075884641392, 4.75590257358561093968103555464, 6.46457894884397639019288564850, 7.30325474886850083987858982335, 7.62312067522231299906329229406, 8.390676973414778789066678399284, 8.799485799941517177228443933912

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