Properties

Label 2-6034-1.1-c1-0-96
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 − 0.368·3-s + 4-s + 1.13·5-s − 0.368·6-s + 7-s + 8-s − 2.86·9-s + 1.13·10-s + 5.79·11-s − 0.368·12-s + 2.85·13-s + 14-s − 0.417·15-s + 16-s − 1.34·17-s − 2.86·18-s − 0.485·19-s + 1.13·20-s − 0.368·21-s + 5.79·22-s + 4.48·23-s − 0.368·24-s − 3.71·25-s + 2.85·26-s + 2.15·27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.212·3-s + 0.5·4-s + 0.507·5-s − 0.150·6-s + 0.377·7-s + 0.353·8-s − 0.954·9-s + 0.358·10-s + 1.74·11-s − 0.106·12-s + 0.792·13-s + 0.267·14-s − 0.107·15-s + 0.250·16-s − 0.326·17-s − 0.675·18-s − 0.111·19-s + 0.253·20-s − 0.0803·21-s + 1.23·22-s + 0.935·23-s − 0.0751·24-s − 0.742·25-s + 0.560·26-s + 0.415·27-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

\(L(1)\) \(\approx\) \(3.880496354\)
\(L(\frac12)\) \(\approx\) \(3.880496354\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.368T + 3T^{2} \)
5 \( 1 - 1.13T + 5T^{2} \)
11 \( 1 - 5.79T + 11T^{2} \)
13 \( 1 - 2.85T + 13T^{2} \)
17 \( 1 + 1.34T + 17T^{2} \)
19 \( 1 + 0.485T + 19T^{2} \)
23 \( 1 - 4.48T + 23T^{2} \)
29 \( 1 - 1.70T + 29T^{2} \)
31 \( 1 - 7.42T + 31T^{2} \)
37 \( 1 + 9.37T + 37T^{2} \)
41 \( 1 - 2.02T + 41T^{2} \)
43 \( 1 + 4.49T + 43T^{2} \)
47 \( 1 - 3.97T + 47T^{2} \)
53 \( 1 - 3.77T + 53T^{2} \)
59 \( 1 + 3.81T + 59T^{2} \)
61 \( 1 - 12.6T + 61T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 - 2.75T + 71T^{2} \)
73 \( 1 + 4.42T + 73T^{2} \)
79 \( 1 + 2.93T + 79T^{2} \)
83 \( 1 + 5.83T + 83T^{2} \)
89 \( 1 + 18.6T + 89T^{2} \)
97 \( 1 + 3.76T + 97T^{2} \)
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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

−8.273909840970602338445561518035, −6.92238545508101393505610402892, −6.63876237819816050916577129928, −5.85381270083623309582421247698, −5.33964468888690311444427777718, −4.40095614172252195594958306930, −3.74549989890818955727379916537, −2.89342266946666011656836467189, −1.88483631150594435304333580672, −1.00313638889665885550432846737, 1.00313638889665885550432846737, 1.88483631150594435304333580672, 2.89342266946666011656836467189, 3.74549989890818955727379916537, 4.40095614172252195594958306930, 5.33964468888690311444427777718, 5.85381270083623309582421247698, 6.63876237819816050916577129928, 6.92238545508101393505610402892, 8.273909840970602338445561518035

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