Properties

Label 2-6010-1.1-c1-0-31
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 1.21·3-s + 4-s + 5-s + 1.21·6-s + 2.32·7-s − 8-s − 1.53·9-s − 10-s + 2.81·11-s − 1.21·12-s − 5.17·13-s − 2.32·14-s − 1.21·15-s + 16-s + 2.75·17-s + 1.53·18-s − 2.93·19-s + 20-s − 2.82·21-s − 2.81·22-s − 2.72·23-s + 1.21·24-s + 25-s + 5.17·26-s + 5.49·27-s + 2.32·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.699·3-s + 0.5·4-s + 0.447·5-s + 0.494·6-s + 0.879·7-s − 0.353·8-s − 0.510·9-s − 0.316·10-s + 0.848·11-s − 0.349·12-s − 1.43·13-s − 0.622·14-s − 0.312·15-s + 0.250·16-s + 0.668·17-s + 0.360·18-s − 0.672·19-s + 0.223·20-s − 0.615·21-s − 0.599·22-s − 0.567·23-s + 0.247·24-s + 0.200·25-s + 1.01·26-s + 1.05·27-s + 0.439·28-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: \(0\)
Selberg data: \((2,\ 6010,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.045872118\)
\(L(\frac12)\) \(\approx\) \(1.045872118\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.21T + 3T^{2} \)
7 \( 1 - 2.32T + 7T^{2} \)
11 \( 1 - 2.81T + 11T^{2} \)
13 \( 1 + 5.17T + 13T^{2} \)
17 \( 1 - 2.75T + 17T^{2} \)
19 \( 1 + 2.93T + 19T^{2} \)
23 \( 1 + 2.72T + 23T^{2} \)
29 \( 1 - 1.77T + 29T^{2} \)
31 \( 1 + 2.88T + 31T^{2} \)
37 \( 1 - 7.74T + 37T^{2} \)
41 \( 1 + 9.53T + 41T^{2} \)
43 \( 1 + 1.99T + 43T^{2} \)
47 \( 1 + 0.274T + 47T^{2} \)
53 \( 1 + 8.62T + 53T^{2} \)
59 \( 1 - 4.64T + 59T^{2} \)
61 \( 1 - 13.5T + 61T^{2} \)
67 \( 1 + 0.401T + 67T^{2} \)
71 \( 1 - 5.59T + 71T^{2} \)
73 \( 1 - 2.36T + 73T^{2} \)
79 \( 1 - 8.10T + 79T^{2} \)
83 \( 1 + 3.51T + 83T^{2} \)
89 \( 1 - 15.3T + 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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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

−8.125705996349153553073328927649, −7.44620936500674492667742679661, −6.59047934335429804831244027296, −6.11141014052475284603473772969, −5.18260657864378479594612294768, −4.77517668877579709199817051424, −3.58208884034090736053074741961, −2.45629891969249926478541853490, −1.73930011057131881955149707682, −0.62012192981807069857747946918, 0.62012192981807069857747946918, 1.73930011057131881955149707682, 2.45629891969249926478541853490, 3.58208884034090736053074741961, 4.77517668877579709199817051424, 5.18260657864378479594612294768, 6.11141014052475284603473772969, 6.59047934335429804831244027296, 7.44620936500674492667742679661, 8.125705996349153553073328927649

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