Properties

Label 2-2450-1.1-c1-0-53
Degree $2$
Conductor $2450$
Sign $-1$
Analytic cond. $19.5633$
Root an. cond. $4.42304$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.41·3-s + 4-s − 1.41·6-s − 8-s − 0.999·9-s − 2·11-s + 1.41·12-s + 16-s + 1.41·17-s + 0.999·18-s − 7.07·19-s + 2·22-s + 4·23-s − 1.41·24-s − 5.65·27-s + 2·29-s + 8.48·31-s − 32-s − 2.82·33-s − 1.41·34-s − 0.999·36-s − 10·37-s + 7.07·38-s − 9.89·41-s − 2·43-s − 2·44-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.816·3-s + 0.5·4-s − 0.577·6-s − 0.353·8-s − 0.333·9-s − 0.603·11-s + 0.408·12-s + 0.250·16-s + 0.342·17-s + 0.235·18-s − 1.62·19-s + 0.426·22-s + 0.834·23-s − 0.288·24-s − 1.08·27-s + 0.371·29-s + 1.52·31-s − 0.176·32-s − 0.492·33-s − 0.242·34-s − 0.166·36-s − 1.64·37-s + 1.14·38-s − 1.54·41-s − 0.304·43-s − 0.301·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2450\)    =    \(2 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(19.5633\)
Root analytic conductor: \(4.42304\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2450,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 - 1.41T + 3T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 1.41T + 17T^{2} \)
19 \( 1 + 7.07T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 8.48T + 31T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 + 9.89T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + 1.41T + 59T^{2} \)
61 \( 1 - 2.82T + 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 1.41T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 9.89T + 83T^{2} \)
89 \( 1 + 7.07T + 89T^{2} \)
97 \( 1 + 9.89T + 97T^{2} \)
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

−8.515948099688136973439282906171, −8.142749811253533413828290341214, −7.17201756060828013185414295338, −6.45997921700750033789536909292, −5.49870623595802656801151655402, −4.50446129371836406425191915992, −3.29999362433018355676864180741, −2.65325475390506882403929399354, −1.64863925254873572140399482410, 0, 1.64863925254873572140399482410, 2.65325475390506882403929399354, 3.29999362433018355676864180741, 4.50446129371836406425191915992, 5.49870623595802656801151655402, 6.45997921700750033789536909292, 7.17201756060828013185414295338, 8.142749811253533413828290341214, 8.515948099688136973439282906171

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