Properties

Label 2-1078-1.1-c1-0-26
Degree $2$
Conductor $1078$
Sign $-1$
Analytic cond. $8.60787$
Root an. cond. $2.93391$
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 − 2.41·3-s + 4-s − 0.585·5-s − 2.41·6-s + 8-s + 2.82·9-s − 0.585·10-s + 11-s − 2.41·12-s − 3.82·13-s + 1.41·15-s + 16-s + 3.65·17-s + 2.82·18-s − 0.585·19-s − 0.585·20-s + 22-s − 6.24·23-s − 2.41·24-s − 4.65·25-s − 3.82·26-s + 0.414·27-s + 2.65·29-s + 1.41·30-s − 4·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.39·3-s + 0.5·4-s − 0.261·5-s − 0.985·6-s + 0.353·8-s + 0.942·9-s − 0.185·10-s + 0.301·11-s − 0.696·12-s − 1.06·13-s + 0.365·15-s + 0.250·16-s + 0.886·17-s + 0.666·18-s − 0.134·19-s − 0.130·20-s + 0.213·22-s − 1.30·23-s − 0.492·24-s − 0.931·25-s − 0.750·26-s + 0.0797·27-s + 0.493·29-s + 0.258·30-s − 0.718·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 \)
11 \( 1 - T \)
good3 \( 1 + 2.41T + 3T^{2} \)
5 \( 1 + 0.585T + 5T^{2} \)
13 \( 1 + 3.82T + 13T^{2} \)
17 \( 1 - 3.65T + 17T^{2} \)
19 \( 1 + 0.585T + 19T^{2} \)
23 \( 1 + 6.24T + 23T^{2} \)
29 \( 1 - 2.65T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 9.41T + 37T^{2} \)
41 \( 1 + 5.41T + 41T^{2} \)
43 \( 1 + 5.65T + 43T^{2} \)
47 \( 1 + 10.4T + 47T^{2} \)
53 \( 1 - 7.89T + 53T^{2} \)
59 \( 1 + 5.58T + 59T^{2} \)
61 \( 1 - 11.8T + 61T^{2} \)
67 \( 1 - 2.75T + 67T^{2} \)
71 \( 1 + 11.0T + 71T^{2} \)
73 \( 1 + 9.41T + 73T^{2} \)
79 \( 1 + 13.2T + 79T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 - 12.4T + 89T^{2} \)
97 \( 1 + 3.82T + 97T^{2} \)
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

−10.06477862546213881074250399468, −8.509975251680844266118807433803, −7.46933604672738163711201981400, −6.75197710276730297124267104510, −5.85494737514462564543108724177, −5.24453842468061287533273340138, −4.40137796680275312007265807935, −3.35509225231927757396983707596, −1.78873006098545990279967493870, 0, 1.78873006098545990279967493870, 3.35509225231927757396983707596, 4.40137796680275312007265807935, 5.24453842468061287533273340138, 5.85494737514462564543108724177, 6.75197710276730297124267104510, 7.46933604672738163711201981400, 8.509975251680844266118807433803, 10.06477862546213881074250399468

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