Properties

Label 2-245-1.1-c3-0-12
Degree $2$
Conductor $245$
Sign $1$
Analytic cond. $14.4554$
Root an. cond. $3.80203$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.227·2-s + 1.80·3-s − 7.94·4-s + 5·5-s − 0.411·6-s + 3.62·8-s − 23.7·9-s − 1.13·10-s + 17.7·11-s − 14.3·12-s + 62.3·13-s + 9.04·15-s + 62.7·16-s − 87.1·17-s + 5.39·18-s + 101.·19-s − 39.7·20-s − 4.02·22-s + 93.6·23-s + 6.56·24-s + 25·25-s − 14.1·26-s − 91.7·27-s + 297.·29-s − 2.05·30-s − 91.2·31-s − 43.3·32-s + ⋯
L(s)  = 1  − 0.0804·2-s + 0.348·3-s − 0.993·4-s + 0.447·5-s − 0.0279·6-s + 0.160·8-s − 0.878·9-s − 0.0359·10-s + 0.485·11-s − 0.345·12-s + 1.32·13-s + 0.155·15-s + 0.980·16-s − 1.24·17-s + 0.0706·18-s + 1.22·19-s − 0.444·20-s − 0.0390·22-s + 0.848·23-s + 0.0558·24-s + 0.200·25-s − 0.106·26-s − 0.653·27-s + 1.90·29-s − 0.0125·30-s − 0.528·31-s − 0.239·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(14.4554\)
Root analytic conductor: \(3.80203\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.668245984\)
\(L(\frac12)\) \(\approx\) \(1.668245984\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 5T \)
7 \( 1 \)
good2 \( 1 + 0.227T + 8T^{2} \)
3 \( 1 - 1.80T + 27T^{2} \)
11 \( 1 - 17.7T + 1.33e3T^{2} \)
13 \( 1 - 62.3T + 2.19e3T^{2} \)
17 \( 1 + 87.1T + 4.91e3T^{2} \)
19 \( 1 - 101.T + 6.85e3T^{2} \)
23 \( 1 - 93.6T + 1.21e4T^{2} \)
29 \( 1 - 297.T + 2.43e4T^{2} \)
31 \( 1 + 91.2T + 2.97e4T^{2} \)
37 \( 1 - 281.T + 5.06e4T^{2} \)
41 \( 1 - 271.T + 6.89e4T^{2} \)
43 \( 1 + 7.81T + 7.95e4T^{2} \)
47 \( 1 - 92.4T + 1.03e5T^{2} \)
53 \( 1 - 180.T + 1.48e5T^{2} \)
59 \( 1 - 99.7T + 2.05e5T^{2} \)
61 \( 1 + 434.T + 2.26e5T^{2} \)
67 \( 1 + 461.T + 3.00e5T^{2} \)
71 \( 1 - 518.T + 3.57e5T^{2} \)
73 \( 1 - 542.T + 3.89e5T^{2} \)
79 \( 1 - 239.T + 4.93e5T^{2} \)
83 \( 1 + 299.T + 5.71e5T^{2} \)
89 \( 1 + 1.05e3T + 7.04e5T^{2} \)
97 \( 1 + 288.T + 9.12e5T^{2} \)
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

−11.55870610075027278007177084818, −10.66802170695124585970897063502, −9.352900985337090351715453840217, −8.917065011371538015840221528622, −8.026854455753620119121632342805, −6.48398662305985046360426472939, −5.44767441737795449699207517782, −4.19909864996661222570005645748, −2.93191103242851845919281771679, −1.00044077987137896975236472421, 1.00044077987137896975236472421, 2.93191103242851845919281771679, 4.19909864996661222570005645748, 5.44767441737795449699207517782, 6.48398662305985046360426472939, 8.026854455753620119121632342805, 8.917065011371538015840221528622, 9.352900985337090351715453840217, 10.66802170695124585970897063502, 11.55870610075027278007177084818

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