Properties

Label 2-6034-1.1-c1-0-32
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 + 1.58·3-s + 4-s − 2.70·5-s − 1.58·6-s + 7-s − 8-s − 0.493·9-s + 2.70·10-s − 3.32·11-s + 1.58·12-s − 0.521·13-s − 14-s − 4.27·15-s + 16-s + 5.72·17-s + 0.493·18-s + 1.35·19-s − 2.70·20-s + 1.58·21-s + 3.32·22-s + 0.952·23-s − 1.58·24-s + 2.30·25-s + 0.521·26-s − 5.53·27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.914·3-s + 0.5·4-s − 1.20·5-s − 0.646·6-s + 0.377·7-s − 0.353·8-s − 0.164·9-s + 0.854·10-s − 1.00·11-s + 0.457·12-s − 0.144·13-s − 0.267·14-s − 1.10·15-s + 0.250·16-s + 1.38·17-s + 0.116·18-s + 0.311·19-s − 0.604·20-s + 0.345·21-s + 0.708·22-s + 0.198·23-s − 0.323·24-s + 0.460·25-s + 0.102·26-s − 1.06·27-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

\(L(1)\) \(\approx\) \(1.185808111\)
\(L(\frac12)\) \(\approx\) \(1.185808111\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.58T + 3T^{2} \)
5 \( 1 + 2.70T + 5T^{2} \)
11 \( 1 + 3.32T + 11T^{2} \)
13 \( 1 + 0.521T + 13T^{2} \)
17 \( 1 - 5.72T + 17T^{2} \)
19 \( 1 - 1.35T + 19T^{2} \)
23 \( 1 - 0.952T + 23T^{2} \)
29 \( 1 - 1.32T + 29T^{2} \)
31 \( 1 + 5.57T + 31T^{2} \)
37 \( 1 + 0.895T + 37T^{2} \)
41 \( 1 - 8.53T + 41T^{2} \)
43 \( 1 + 5.56T + 43T^{2} \)
47 \( 1 - 11.8T + 47T^{2} \)
53 \( 1 + 9.76T + 53T^{2} \)
59 \( 1 + 0.914T + 59T^{2} \)
61 \( 1 + 2.32T + 61T^{2} \)
67 \( 1 + 4.33T + 67T^{2} \)
71 \( 1 - 5.95T + 71T^{2} \)
73 \( 1 + 5.02T + 73T^{2} \)
79 \( 1 - 0.371T + 79T^{2} \)
83 \( 1 - 6.44T + 83T^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 - 7.32T + 97T^{2} \)
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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

−8.031965311843290808499860545306, −7.59631797617446937067810124981, −7.25035075483357765880208223988, −5.92090146320424831716452300807, −5.26020590843862247321496138981, −4.24330170367264792317037675053, −3.35278633057662975517992249521, −2.89016806522973971402369010887, −1.87231099767934115351034939466, −0.59248506103041994353277359997, 0.59248506103041994353277359997, 1.87231099767934115351034939466, 2.89016806522973971402369010887, 3.35278633057662975517992249521, 4.24330170367264792317037675053, 5.26020590843862247321496138981, 5.92090146320424831716452300807, 7.25035075483357765880208223988, 7.59631797617446937067810124981, 8.031965311843290808499860545306

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