Properties

Label 2-6010-1.1-c1-0-164
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 − 0.310·3-s + 4-s − 5-s − 0.310·6-s + 1.47·7-s + 8-s − 2.90·9-s − 10-s + 2.97·11-s − 0.310·12-s − 6.20·13-s + 1.47·14-s + 0.310·15-s + 16-s − 4.41·17-s − 2.90·18-s + 4.87·19-s − 20-s − 0.459·21-s + 2.97·22-s + 2.42·23-s − 0.310·24-s + 25-s − 6.20·26-s + 1.83·27-s + 1.47·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.179·3-s + 0.5·4-s − 0.447·5-s − 0.126·6-s + 0.559·7-s + 0.353·8-s − 0.967·9-s − 0.316·10-s + 0.896·11-s − 0.0897·12-s − 1.72·13-s + 0.395·14-s + 0.0802·15-s + 0.250·16-s − 1.07·17-s − 0.684·18-s + 1.11·19-s − 0.223·20-s − 0.100·21-s + 0.634·22-s + 0.504·23-s − 0.0634·24-s + 0.200·25-s − 1.21·26-s + 0.353·27-s + 0.279·28-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 \( 1 + 0.310T + 3T^{2} \)
7 \( 1 - 1.47T + 7T^{2} \)
11 \( 1 - 2.97T + 11T^{2} \)
13 \( 1 + 6.20T + 13T^{2} \)
17 \( 1 + 4.41T + 17T^{2} \)
19 \( 1 - 4.87T + 19T^{2} \)
23 \( 1 - 2.42T + 23T^{2} \)
29 \( 1 - 2.62T + 29T^{2} \)
31 \( 1 + 3.97T + 31T^{2} \)
37 \( 1 - 4.24T + 37T^{2} \)
41 \( 1 - 10.8T + 41T^{2} \)
43 \( 1 + 4.53T + 43T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 + 9.83T + 53T^{2} \)
59 \( 1 + 6.88T + 59T^{2} \)
61 \( 1 - 4.95T + 61T^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 + 2.88T + 71T^{2} \)
73 \( 1 + 7.23T + 73T^{2} \)
79 \( 1 - 9.98T + 79T^{2} \)
83 \( 1 - 0.667T + 83T^{2} \)
89 \( 1 + 9.17T + 89T^{2} \)
97 \( 1 + 5.27T + 97T^{2} \)
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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

−7.65897156499969744245859378776, −6.92750909751821447903503166311, −6.31175726067322915028664941720, −5.38045484322839951928535880678, −4.81513009874036536857795778553, −4.25987462890362308784205152937, −3.16416876768140210225440153566, −2.58177229269746697563730742907, −1.45004477360020146825886176944, 0, 1.45004477360020146825886176944, 2.58177229269746697563730742907, 3.16416876768140210225440153566, 4.25987462890362308784205152937, 4.81513009874036536857795778553, 5.38045484322839951928535880678, 6.31175726067322915028664941720, 6.92750909751821447903503166311, 7.65897156499969744245859378776

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