Properties

Label 2-6010-1.1-c1-0-195
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 + 2.14·3-s + 4-s − 5-s + 2.14·6-s − 1.75·7-s + 8-s + 1.61·9-s − 10-s + 0.540·11-s + 2.14·12-s + 2.12·13-s − 1.75·14-s − 2.14·15-s + 16-s − 7.79·17-s + 1.61·18-s − 3.30·19-s − 20-s − 3.76·21-s + 0.540·22-s − 2.89·23-s + 2.14·24-s + 25-s + 2.12·26-s − 2.98·27-s − 1.75·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.23·3-s + 0.5·4-s − 0.447·5-s + 0.876·6-s − 0.662·7-s + 0.353·8-s + 0.537·9-s − 0.316·10-s + 0.163·11-s + 0.619·12-s + 0.588·13-s − 0.468·14-s − 0.554·15-s + 0.250·16-s − 1.88·17-s + 0.379·18-s − 0.757·19-s − 0.223·20-s − 0.821·21-s + 0.115·22-s − 0.604·23-s + 0.438·24-s + 0.200·25-s + 0.415·26-s − 0.573·27-s − 0.331·28-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 \( 1 - 2.14T + 3T^{2} \)
7 \( 1 + 1.75T + 7T^{2} \)
11 \( 1 - 0.540T + 11T^{2} \)
13 \( 1 - 2.12T + 13T^{2} \)
17 \( 1 + 7.79T + 17T^{2} \)
19 \( 1 + 3.30T + 19T^{2} \)
23 \( 1 + 2.89T + 23T^{2} \)
29 \( 1 + 4.12T + 29T^{2} \)
31 \( 1 + 10.4T + 31T^{2} \)
37 \( 1 + 2.45T + 37T^{2} \)
41 \( 1 - 9.88T + 41T^{2} \)
43 \( 1 - 3.10T + 43T^{2} \)
47 \( 1 + 5.83T + 47T^{2} \)
53 \( 1 + 6.93T + 53T^{2} \)
59 \( 1 - 13.6T + 59T^{2} \)
61 \( 1 + 3.96T + 61T^{2} \)
67 \( 1 - 6.95T + 67T^{2} \)
71 \( 1 + 14.0T + 71T^{2} \)
73 \( 1 - 6.47T + 73T^{2} \)
79 \( 1 - 13.9T + 79T^{2} \)
83 \( 1 + 6.51T + 83T^{2} \)
89 \( 1 - 1.57T + 89T^{2} \)
97 \( 1 - 1.87T + 97T^{2} \)
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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

−7.75698981271390825805564315985, −6.97830378920390167900357080561, −6.38670993953311342382947111946, −5.62228049504828027091635902871, −4.48925017579568498990322612765, −3.86980766398799351177713562442, −3.41505099674698248214880300327, −2.43432131095852972895064239816, −1.85915465826687288853091639567, 0, 1.85915465826687288853091639567, 2.43432131095852972895064239816, 3.41505099674698248214880300327, 3.86980766398799351177713562442, 4.48925017579568498990322612765, 5.62228049504828027091635902871, 6.38670993953311342382947111946, 6.97830378920390167900357080561, 7.75698981271390825805564315985

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