Properties

Label 2-6014-1.1-c1-0-7
Degree $2$
Conductor $6014$
Sign $1$
Analytic cond. $48.0220$
Root an. cond. $6.92979$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3.21·3-s + 4-s − 0.168·5-s + 3.21·6-s + 2.46·7-s − 8-s + 7.32·9-s + 0.168·10-s − 1.60·11-s − 3.21·12-s − 3.41·13-s − 2.46·14-s + 0.540·15-s + 16-s − 7.07·17-s − 7.32·18-s − 4.86·19-s − 0.168·20-s − 7.91·21-s + 1.60·22-s − 8.15·23-s + 3.21·24-s − 4.97·25-s + 3.41·26-s − 13.8·27-s + 2.46·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.85·3-s + 0.5·4-s − 0.0752·5-s + 1.31·6-s + 0.931·7-s − 0.353·8-s + 2.44·9-s + 0.0532·10-s − 0.483·11-s − 0.927·12-s − 0.948·13-s − 0.658·14-s + 0.139·15-s + 0.250·16-s − 1.71·17-s − 1.72·18-s − 1.11·19-s − 0.0376·20-s − 1.72·21-s + 0.341·22-s − 1.69·23-s + 0.655·24-s − 0.994·25-s + 0.670·26-s − 2.67·27-s + 0.465·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6014\)    =    \(2 \cdot 31 \cdot 97\)
Sign: $1$
Analytic conductor: \(48.0220\)
Root analytic conductor: \(6.92979\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6014,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1045844982\)
\(L(\frac12)\) \(\approx\) \(0.1045844982\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
31 \( 1 - T \)
97 \( 1 + T \)
good3 \( 1 + 3.21T + 3T^{2} \)
5 \( 1 + 0.168T + 5T^{2} \)
7 \( 1 - 2.46T + 7T^{2} \)
11 \( 1 + 1.60T + 11T^{2} \)
13 \( 1 + 3.41T + 13T^{2} \)
17 \( 1 + 7.07T + 17T^{2} \)
19 \( 1 + 4.86T + 19T^{2} \)
23 \( 1 + 8.15T + 23T^{2} \)
29 \( 1 + 0.180T + 29T^{2} \)
37 \( 1 + 5.75T + 37T^{2} \)
41 \( 1 - 2.81T + 41T^{2} \)
43 \( 1 + 8.62T + 43T^{2} \)
47 \( 1 + 1.16T + 47T^{2} \)
53 \( 1 - 0.317T + 53T^{2} \)
59 \( 1 - 4.26T + 59T^{2} \)
61 \( 1 - 7.59T + 61T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 + 2.37T + 71T^{2} \)
73 \( 1 - 1.68T + 73T^{2} \)
79 \( 1 - 1.41T + 79T^{2} \)
83 \( 1 + 3.80T + 83T^{2} \)
89 \( 1 + 0.520T + 89T^{2} \)
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

−8.004836563559368908646830939213, −7.28819744822080752133340113492, −6.62241664065840526334462584115, −6.05430684627859346228759470164, −5.26954729269719543668535935577, −4.61687802296450077218422758964, −4.04407838467898338847975973390, −2.21456421952608699418550676233, −1.72740080021219281431815174392, −0.20145900537793572633530097645, 0.20145900537793572633530097645, 1.72740080021219281431815174392, 2.21456421952608699418550676233, 4.04407838467898338847975973390, 4.61687802296450077218422758964, 5.26954729269719543668535935577, 6.05430684627859346228759470164, 6.62241664065840526334462584115, 7.28819744822080752133340113492, 8.004836563559368908646830939213

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