Properties

Label 2-6014-1.1-c1-0-69
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2.07·3-s + 4-s − 2.52·5-s + 2.07·6-s − 2.03·7-s + 8-s + 1.30·9-s − 2.52·10-s − 3.55·11-s + 2.07·12-s − 0.632·13-s − 2.03·14-s − 5.23·15-s + 16-s + 0.247·17-s + 1.30·18-s + 5.31·19-s − 2.52·20-s − 4.22·21-s − 3.55·22-s + 6.96·23-s + 2.07·24-s + 1.35·25-s − 0.632·26-s − 3.51·27-s − 2.03·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.19·3-s + 0.5·4-s − 1.12·5-s + 0.847·6-s − 0.768·7-s + 0.353·8-s + 0.436·9-s − 0.797·10-s − 1.07·11-s + 0.599·12-s − 0.175·13-s − 0.543·14-s − 1.35·15-s + 0.250·16-s + 0.0599·17-s + 0.308·18-s + 1.21·19-s − 0.563·20-s − 0.920·21-s − 0.757·22-s + 1.45·23-s + 0.423·24-s + 0.271·25-s − 0.124·26-s − 0.675·27-s − 0.384·28-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

\(L(1)\) \(\approx\) \(3.263575520\)
\(L(\frac12)\) \(\approx\) \(3.263575520\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.07T + 3T^{2} \)
5 \( 1 + 2.52T + 5T^{2} \)
7 \( 1 + 2.03T + 7T^{2} \)
11 \( 1 + 3.55T + 11T^{2} \)
13 \( 1 + 0.632T + 13T^{2} \)
17 \( 1 - 0.247T + 17T^{2} \)
19 \( 1 - 5.31T + 19T^{2} \)
23 \( 1 - 6.96T + 23T^{2} \)
29 \( 1 + 1.13T + 29T^{2} \)
37 \( 1 - 7.36T + 37T^{2} \)
41 \( 1 - 4.30T + 41T^{2} \)
43 \( 1 - 11.6T + 43T^{2} \)
47 \( 1 - 2.30T + 47T^{2} \)
53 \( 1 - 4.11T + 53T^{2} \)
59 \( 1 - 10.5T + 59T^{2} \)
61 \( 1 - 7.01T + 61T^{2} \)
67 \( 1 + 4.54T + 67T^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 - 13.8T + 73T^{2} \)
79 \( 1 - 16.7T + 79T^{2} \)
83 \( 1 + 2.36T + 83T^{2} \)
89 \( 1 + 0.486T + 89T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.78208469372438145468329887005, −7.59523638463737779732673117448, −6.88556070222001513710006281179, −5.80911517923404093981179157417, −5.12093042851545914193663931848, −4.17688428394027166831860508865, −3.55631942708610025977338725366, −2.88339988788862973783763865643, −2.45680529416211907758612422641, −0.78099081890565181942381967174, 0.78099081890565181942381967174, 2.45680529416211907758612422641, 2.88339988788862973783763865643, 3.55631942708610025977338725366, 4.17688428394027166831860508865, 5.12093042851545914193663931848, 5.80911517923404093981179157417, 6.88556070222001513710006281179, 7.59523638463737779732673117448, 7.78208469372438145468329887005

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