Properties

Label 2-6014-1.1-c1-0-158
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 + 2.23·3-s + 4-s + 4.33·5-s − 2.23·6-s + 3.84·7-s − 8-s + 1.99·9-s − 4.33·10-s + 0.985·11-s + 2.23·12-s − 5.10·13-s − 3.84·14-s + 9.68·15-s + 16-s + 3.80·17-s − 1.99·18-s − 2.32·19-s + 4.33·20-s + 8.58·21-s − 0.985·22-s + 6.44·23-s − 2.23·24-s + 13.7·25-s + 5.10·26-s − 2.23·27-s + 3.84·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.29·3-s + 0.5·4-s + 1.93·5-s − 0.912·6-s + 1.45·7-s − 0.353·8-s + 0.666·9-s − 1.36·10-s + 0.297·11-s + 0.645·12-s − 1.41·13-s − 1.02·14-s + 2.49·15-s + 0.250·16-s + 0.923·17-s − 0.471·18-s − 0.532·19-s + 0.968·20-s + 1.87·21-s − 0.210·22-s + 1.34·23-s − 0.456·24-s + 2.75·25-s + 1.00·26-s − 0.430·27-s + 0.725·28-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

\(L(1)\) \(\approx\) \(4.154253396\)
\(L(\frac12)\) \(\approx\) \(4.154253396\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23T + 3T^{2} \)
5 \( 1 - 4.33T + 5T^{2} \)
7 \( 1 - 3.84T + 7T^{2} \)
11 \( 1 - 0.985T + 11T^{2} \)
13 \( 1 + 5.10T + 13T^{2} \)
17 \( 1 - 3.80T + 17T^{2} \)
19 \( 1 + 2.32T + 19T^{2} \)
23 \( 1 - 6.44T + 23T^{2} \)
29 \( 1 + 2.94T + 29T^{2} \)
37 \( 1 - 2.66T + 37T^{2} \)
41 \( 1 + 7.12T + 41T^{2} \)
43 \( 1 + 8.08T + 43T^{2} \)
47 \( 1 - 12.2T + 47T^{2} \)
53 \( 1 + 7.50T + 53T^{2} \)
59 \( 1 - 1.69T + 59T^{2} \)
61 \( 1 - 6.35T + 61T^{2} \)
67 \( 1 + 12.8T + 67T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 + 10.9T + 73T^{2} \)
79 \( 1 + 9.16T + 79T^{2} \)
83 \( 1 + 6.75T + 83T^{2} \)
89 \( 1 + 1.26T + 89T^{2} \)
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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

−8.361818809407391568246671366207, −7.42911078832195126395522073882, −7.01746159655940008529220213357, −5.92254338406956950897284899727, −5.24792958099871962131745724174, −4.60757772549431037857391423168, −3.17802720863603767470020263592, −2.47709449841744746010852108818, −1.89144847306440216684034895155, −1.26932283895849158306296474051, 1.26932283895849158306296474051, 1.89144847306440216684034895155, 2.47709449841744746010852108818, 3.17802720863603767470020263592, 4.60757772549431037857391423168, 5.24792958099871962131745724174, 5.92254338406956950897284899727, 7.01746159655940008529220213357, 7.42911078832195126395522073882, 8.361818809407391568246671366207

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