Properties

Label 2-6014-1.1-c1-0-233
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 $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 1.71·3-s + 4-s + 2.08·5-s + 1.71·6-s − 3.84·7-s + 8-s − 0.0606·9-s + 2.08·10-s − 4.60·11-s + 1.71·12-s + 1.14·13-s − 3.84·14-s + 3.57·15-s + 16-s − 3.79·17-s − 0.0606·18-s − 2.00·19-s + 2.08·20-s − 6.59·21-s − 4.60·22-s + 1.16·23-s + 1.71·24-s − 0.646·25-s + 1.14·26-s − 5.24·27-s − 3.84·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.989·3-s + 0.5·4-s + 0.933·5-s + 0.699·6-s − 1.45·7-s + 0.353·8-s − 0.0202·9-s + 0.659·10-s − 1.38·11-s + 0.494·12-s + 0.316·13-s − 1.02·14-s + 0.923·15-s + 0.250·16-s − 0.921·17-s − 0.0142·18-s − 0.460·19-s + 0.466·20-s − 1.43·21-s − 0.981·22-s + 0.242·23-s + 0.349·24-s − 0.129·25-s + 0.223·26-s − 1.00·27-s − 0.727·28-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: \(1\)
Selberg data: \((2,\ 6014,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
31 \( 1 - T \)
97 \( 1 + T \)
good3 \( 1 - 1.71T + 3T^{2} \)
5 \( 1 - 2.08T + 5T^{2} \)
7 \( 1 + 3.84T + 7T^{2} \)
11 \( 1 + 4.60T + 11T^{2} \)
13 \( 1 - 1.14T + 13T^{2} \)
17 \( 1 + 3.79T + 17T^{2} \)
19 \( 1 + 2.00T + 19T^{2} \)
23 \( 1 - 1.16T + 23T^{2} \)
29 \( 1 - 0.335T + 29T^{2} \)
37 \( 1 - 2.98T + 37T^{2} \)
41 \( 1 - 1.97T + 41T^{2} \)
43 \( 1 + 6.12T + 43T^{2} \)
47 \( 1 + 1.50T + 47T^{2} \)
53 \( 1 + 1.84T + 53T^{2} \)
59 \( 1 + 7.71T + 59T^{2} \)
61 \( 1 - 11.0T + 61T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + 2.82T + 71T^{2} \)
73 \( 1 + 11.7T + 73T^{2} \)
79 \( 1 - 2.36T + 79T^{2} \)
83 \( 1 + 12.5T + 83T^{2} \)
89 \( 1 - 3.27T + 89T^{2} \)
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

−7.68847730144575393872860431092, −6.89546680022811776059774411534, −6.14436173071530217473928112954, −5.75340077255570691343911511389, −4.80921098736762973442106821618, −3.87407480357628910394491190144, −2.98429445872514786409746572710, −2.66379388721638301686730682362, −1.85185679930903283603916612331, 0, 1.85185679930903283603916612331, 2.66379388721638301686730682362, 2.98429445872514786409746572710, 3.87407480357628910394491190144, 4.80921098736762973442106821618, 5.75340077255570691343911511389, 6.14436173071530217473928112954, 6.89546680022811776059774411534, 7.68847730144575393872860431092

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