Properties

Label 2-6014-1.1-c1-0-66
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 − 2.85·3-s + 4-s + 2.60·5-s − 2.85·6-s − 2.89·7-s + 8-s + 5.17·9-s + 2.60·10-s + 2.99·11-s − 2.85·12-s − 4.16·13-s − 2.89·14-s − 7.43·15-s + 16-s + 2.55·17-s + 5.17·18-s + 4.52·19-s + 2.60·20-s + 8.26·21-s + 2.99·22-s + 2.80·23-s − 2.85·24-s + 1.76·25-s − 4.16·26-s − 6.21·27-s − 2.89·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.65·3-s + 0.5·4-s + 1.16·5-s − 1.16·6-s − 1.09·7-s + 0.353·8-s + 1.72·9-s + 0.822·10-s + 0.902·11-s − 0.825·12-s − 1.15·13-s − 0.772·14-s − 1.91·15-s + 0.250·16-s + 0.620·17-s + 1.21·18-s + 1.03·19-s + 0.581·20-s + 1.80·21-s + 0.638·22-s + 0.585·23-s − 0.583·24-s + 0.352·25-s − 0.816·26-s − 1.19·27-s − 0.546·28-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

\(L(1)\) \(\approx\) \(1.973634191\)
\(L(\frac12)\) \(\approx\) \(1.973634191\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.85T + 3T^{2} \)
5 \( 1 - 2.60T + 5T^{2} \)
7 \( 1 + 2.89T + 7T^{2} \)
11 \( 1 - 2.99T + 11T^{2} \)
13 \( 1 + 4.16T + 13T^{2} \)
17 \( 1 - 2.55T + 17T^{2} \)
19 \( 1 - 4.52T + 19T^{2} \)
23 \( 1 - 2.80T + 23T^{2} \)
29 \( 1 + 3.38T + 29T^{2} \)
37 \( 1 + 1.20T + 37T^{2} \)
41 \( 1 + 4.87T + 41T^{2} \)
43 \( 1 - 4.40T + 43T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 + 1.01T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 + 12.5T + 67T^{2} \)
71 \( 1 - 14.8T + 71T^{2} \)
73 \( 1 + 5.98T + 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 - 8.26T + 83T^{2} \)
89 \( 1 + 6.86T + 89T^{2} \)
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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

−7.61530683661746085503014627772, −6.93733303009595167079648971213, −6.47262002149544497058384296617, −5.83430526212176122433900968932, −5.38350084086590341405436671832, −4.77827038656484457558781080515, −3.76885256926093133481128017127, −2.87095484129621821834404723192, −1.75878242747728436630095739431, −0.73454047081618681510853452639, 0.73454047081618681510853452639, 1.75878242747728436630095739431, 2.87095484129621821834404723192, 3.76885256926093133481128017127, 4.77827038656484457558781080515, 5.38350084086590341405436671832, 5.83430526212176122433900968932, 6.47262002149544497058384296617, 6.93733303009595167079648971213, 7.61530683661746085503014627772

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