Properties

Label 2-6010-1.1-c1-0-154
Degree $2$
Conductor $6010$
Sign $-1$
Analytic cond. $47.9900$
Root an. cond. $6.92748$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2.45·3-s + 4-s + 5-s − 2.45·6-s + 1.53·7-s + 8-s + 3.01·9-s + 10-s − 0.396·11-s − 2.45·12-s − 1.09·13-s + 1.53·14-s − 2.45·15-s + 16-s + 3.63·17-s + 3.01·18-s − 2.31·19-s + 20-s − 3.75·21-s − 0.396·22-s − 7.59·23-s − 2.45·24-s + 25-s − 1.09·26-s − 0.0293·27-s + 1.53·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.41·3-s + 0.5·4-s + 0.447·5-s − 1.00·6-s + 0.578·7-s + 0.353·8-s + 1.00·9-s + 0.316·10-s − 0.119·11-s − 0.707·12-s − 0.303·13-s + 0.409·14-s − 0.633·15-s + 0.250·16-s + 0.882·17-s + 0.709·18-s − 0.530·19-s + 0.223·20-s − 0.819·21-s − 0.0844·22-s − 1.58·23-s − 0.500·24-s + 0.200·25-s − 0.214·26-s − 0.00565·27-s + 0.289·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6010\)    =    \(2 \cdot 5 \cdot 601\)
Sign: $-1$
Analytic conductor: \(47.9900\)
Root analytic conductor: \(6.92748\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6010,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 - T \)
601 \( 1 + T \)
good3 \( 1 + 2.45T + 3T^{2} \)
7 \( 1 - 1.53T + 7T^{2} \)
11 \( 1 + 0.396T + 11T^{2} \)
13 \( 1 + 1.09T + 13T^{2} \)
17 \( 1 - 3.63T + 17T^{2} \)
19 \( 1 + 2.31T + 19T^{2} \)
23 \( 1 + 7.59T + 23T^{2} \)
29 \( 1 + 0.633T + 29T^{2} \)
31 \( 1 + 1.51T + 31T^{2} \)
37 \( 1 + 9.28T + 37T^{2} \)
41 \( 1 + 8.32T + 41T^{2} \)
43 \( 1 - 4.30T + 43T^{2} \)
47 \( 1 - 1.40T + 47T^{2} \)
53 \( 1 + 3.05T + 53T^{2} \)
59 \( 1 + 3.82T + 59T^{2} \)
61 \( 1 + 1.88T + 61T^{2} \)
67 \( 1 - 0.761T + 67T^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 + 9.16T + 73T^{2} \)
79 \( 1 - 12.0T + 79T^{2} \)
83 \( 1 - 5.18T + 83T^{2} \)
89 \( 1 - 4.52T + 89T^{2} \)
97 \( 1 + 3.00T + 97T^{2} \)
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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

−7.58293348278116702737369207464, −6.63384226809793229963317866875, −6.22571901987212194023979731996, −5.36591226517684005881756247539, −5.16079572536917866717205822456, −4.28825875823016408818936206058, −3.42289906896908591853188942910, −2.19409978676937454846516888588, −1.39974209077554728840810092960, 0, 1.39974209077554728840810092960, 2.19409978676937454846516888588, 3.42289906896908591853188942910, 4.28825875823016408818936206058, 5.16079572536917866717205822456, 5.36591226517684005881756247539, 6.22571901987212194023979731996, 6.63384226809793229963317866875, 7.58293348278116702737369207464

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