Properties

Label 2-6034-1.1-c1-0-155
Degree $2$
Conductor $6034$
Sign $-1$
Analytic cond. $48.1817$
Root an. cond. $6.94130$
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 − 1.35·3-s + 4-s + 1.14·5-s − 1.35·6-s − 7-s + 8-s − 1.16·9-s + 1.14·10-s − 4.90·11-s − 1.35·12-s + 3.53·13-s − 14-s − 1.55·15-s + 16-s − 2.10·17-s − 1.16·18-s + 2.64·19-s + 1.14·20-s + 1.35·21-s − 4.90·22-s + 1.73·23-s − 1.35·24-s − 3.68·25-s + 3.53·26-s + 5.64·27-s − 28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.782·3-s + 0.5·4-s + 0.512·5-s − 0.553·6-s − 0.377·7-s + 0.353·8-s − 0.386·9-s + 0.362·10-s − 1.47·11-s − 0.391·12-s + 0.980·13-s − 0.267·14-s − 0.401·15-s + 0.250·16-s − 0.510·17-s − 0.273·18-s + 0.605·19-s + 0.256·20-s + 0.295·21-s − 1.04·22-s + 0.360·23-s − 0.276·24-s − 0.737·25-s + 0.693·26-s + 1.08·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6034\)    =    \(2 \cdot 7 \cdot 431\)
Sign: $-1$
Analytic conductor: \(48.1817\)
Root analytic conductor: \(6.94130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6034,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
431 \( 1 - T \)
good3 \( 1 + 1.35T + 3T^{2} \)
5 \( 1 - 1.14T + 5T^{2} \)
11 \( 1 + 4.90T + 11T^{2} \)
13 \( 1 - 3.53T + 13T^{2} \)
17 \( 1 + 2.10T + 17T^{2} \)
19 \( 1 - 2.64T + 19T^{2} \)
23 \( 1 - 1.73T + 23T^{2} \)
29 \( 1 + 1.92T + 29T^{2} \)
31 \( 1 - 7.14T + 31T^{2} \)
37 \( 1 - 11.0T + 37T^{2} \)
41 \( 1 + 7.53T + 41T^{2} \)
43 \( 1 - 2.28T + 43T^{2} \)
47 \( 1 - 3.23T + 47T^{2} \)
53 \( 1 + 13.5T + 53T^{2} \)
59 \( 1 + 3.53T + 59T^{2} \)
61 \( 1 + 0.0973T + 61T^{2} \)
67 \( 1 + 11.6T + 67T^{2} \)
71 \( 1 + 6.82T + 71T^{2} \)
73 \( 1 + 5.06T + 73T^{2} \)
79 \( 1 + 2.22T + 79T^{2} \)
83 \( 1 + 7.91T + 83T^{2} \)
89 \( 1 - 0.879T + 89T^{2} \)
97 \( 1 + 11.1T + 97T^{2} \)
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

−7.66830272495015724487201640938, −6.68630688472197087639808483749, −6.05199389288550617609441929792, −5.69220564554747261051347489350, −4.97161387144551800877659823884, −4.24437532148533857197564927970, −3.06246449113217523307091358402, −2.62750223451303194719828844997, −1.34800836362455330124231726181, 0, 1.34800836362455330124231726181, 2.62750223451303194719828844997, 3.06246449113217523307091358402, 4.24437532148533857197564927970, 4.97161387144551800877659823884, 5.69220564554747261051347489350, 6.05199389288550617609441929792, 6.68630688472197087639808483749, 7.66830272495015724487201640938

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