Properties

Label 2-6034-1.1-c1-0-1
Degree $2$
Conductor $6034$
Sign $1$
Analytic cond. $48.1817$
Root an. cond. $6.94130$
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 − 1.81·3-s + 4-s − 1.26·5-s + 1.81·6-s + 7-s − 8-s + 0.281·9-s + 1.26·10-s − 3.69·11-s − 1.81·12-s − 1.60·13-s − 14-s + 2.29·15-s + 16-s + 1.22·17-s − 0.281·18-s − 8.48·19-s − 1.26·20-s − 1.81·21-s + 3.69·22-s + 2.94·23-s + 1.81·24-s − 3.40·25-s + 1.60·26-s + 4.92·27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.04·3-s + 0.5·4-s − 0.565·5-s + 0.739·6-s + 0.377·7-s − 0.353·8-s + 0.0938·9-s + 0.399·10-s − 1.11·11-s − 0.522·12-s − 0.446·13-s − 0.267·14-s + 0.591·15-s + 0.250·16-s + 0.297·17-s − 0.0663·18-s − 1.94·19-s − 0.282·20-s − 0.395·21-s + 0.788·22-s + 0.614·23-s + 0.369·24-s − 0.680·25-s + 0.315·26-s + 0.947·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6034\)    =    \(2 \cdot 7 \cdot 431\)
Sign: $1$
Analytic conductor: \(48.1817\)
Root analytic conductor: \(6.94130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6034,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1096232224\)
\(L(\frac12)\) \(\approx\) \(0.1096232224\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 - T \)
431 \( 1 + T \)
good3 \( 1 + 1.81T + 3T^{2} \)
5 \( 1 + 1.26T + 5T^{2} \)
11 \( 1 + 3.69T + 11T^{2} \)
13 \( 1 + 1.60T + 13T^{2} \)
17 \( 1 - 1.22T + 17T^{2} \)
19 \( 1 + 8.48T + 19T^{2} \)
23 \( 1 - 2.94T + 23T^{2} \)
29 \( 1 + 5.38T + 29T^{2} \)
31 \( 1 - 1.88T + 31T^{2} \)
37 \( 1 + 5.44T + 37T^{2} \)
41 \( 1 + 3.98T + 41T^{2} \)
43 \( 1 + 12.2T + 43T^{2} \)
47 \( 1 - 0.841T + 47T^{2} \)
53 \( 1 - 1.85T + 53T^{2} \)
59 \( 1 + 9.72T + 59T^{2} \)
61 \( 1 + 5.36T + 61T^{2} \)
67 \( 1 + 1.54T + 67T^{2} \)
71 \( 1 - 0.705T + 71T^{2} \)
73 \( 1 - 15.3T + 73T^{2} \)
79 \( 1 - 3.15T + 79T^{2} \)
83 \( 1 + 1.30T + 83T^{2} \)
89 \( 1 + 9.94T + 89T^{2} \)
97 \( 1 + 3.24T + 97T^{2} \)
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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

−8.231967700873721856084253276967, −7.38864634060844110472915159331, −6.73082033343481086240819785535, −6.00442308147207399978642551941, −5.22712408727640134857789457088, −4.70287150470550399381538370898, −3.63218169974266134791805568839, −2.59214864599063279606756815330, −1.66474568662357549133347101370, −0.19766236532441481557611082333, 0.19766236532441481557611082333, 1.66474568662357549133347101370, 2.59214864599063279606756815330, 3.63218169974266134791805568839, 4.70287150470550399381538370898, 5.22712408727640134857789457088, 6.00442308147207399978642551941, 6.73082033343481086240819785535, 7.38864634060844110472915159331, 8.231967700873721856084253276967

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