Properties

Label 2-5550-1.1-c1-0-111
Degree $2$
Conductor $5550$
Sign $-1$
Analytic cond. $44.3169$
Root an. cond. $6.65709$
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 + 3-s + 4-s + 6-s + 2.41·7-s + 8-s + 9-s − 6.41·11-s + 12-s − 5.24·13-s + 2.41·14-s + 16-s − 3.58·17-s + 18-s − 2.17·19-s + 2.41·21-s − 6.41·22-s − 3·23-s + 24-s − 5.24·26-s + 27-s + 2.41·28-s + 7.07·29-s − 3.65·31-s + 32-s − 6.41·33-s − 3.58·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.912·7-s + 0.353·8-s + 0.333·9-s − 1.93·11-s + 0.288·12-s − 1.45·13-s + 0.645·14-s + 0.250·16-s − 0.869·17-s + 0.235·18-s − 0.498·19-s + 0.526·21-s − 1.36·22-s − 0.625·23-s + 0.204·24-s − 1.02·26-s + 0.192·27-s + 0.456·28-s + 1.31·29-s − 0.656·31-s + 0.176·32-s − 1.11·33-s − 0.614·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5550\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 37\)
Sign: $-1$
Analytic conductor: \(44.3169\)
Root analytic conductor: \(6.65709\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5550,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 \)
37 \( 1 - T \)
good7 \( 1 - 2.41T + 7T^{2} \)
11 \( 1 + 6.41T + 11T^{2} \)
13 \( 1 + 5.24T + 13T^{2} \)
17 \( 1 + 3.58T + 17T^{2} \)
19 \( 1 + 2.17T + 19T^{2} \)
23 \( 1 + 3T + 23T^{2} \)
29 \( 1 - 7.07T + 29T^{2} \)
31 \( 1 + 3.65T + 31T^{2} \)
41 \( 1 + 6.58T + 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 - 11.8T + 47T^{2} \)
53 \( 1 - 3.82T + 53T^{2} \)
59 \( 1 + 13.0T + 59T^{2} \)
61 \( 1 + 6.48T + 61T^{2} \)
67 \( 1 - 0.828T + 67T^{2} \)
71 \( 1 + 9.41T + 71T^{2} \)
73 \( 1 + 2.17T + 73T^{2} \)
79 \( 1 - 12.2T + 79T^{2} \)
83 \( 1 + 6.89T + 83T^{2} \)
89 \( 1 - 2.89T + 89T^{2} \)
97 \( 1 - 6.72T + 97T^{2} \)
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

−7.75142775167152931174962864583, −7.24379331343296561611113617713, −6.34558239713159400990190468701, −5.31462586621855698817703645934, −4.84380782182716780208563681983, −4.32516918556379983574746213474, −3.09263472157620499758312176020, −2.44043439954830776966115344835, −1.84835139734570715143147531842, 0, 1.84835139734570715143147531842, 2.44043439954830776966115344835, 3.09263472157620499758312176020, 4.32516918556379983574746213474, 4.84380782182716780208563681983, 5.31462586621855698817703645934, 6.34558239713159400990190468701, 7.24379331343296561611113617713, 7.75142775167152931174962864583

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