Properties

Label 2-6014-1.1-c1-0-194
Degree $2$
Conductor $6014$
Sign $1$
Analytic cond. $48.0220$
Root an. cond. $6.92979$
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 + 1.69·3-s + 4-s + 3.58·5-s + 1.69·6-s + 3.25·7-s + 8-s − 0.111·9-s + 3.58·10-s + 3.58·11-s + 1.69·12-s − 5.26·13-s + 3.25·14-s + 6.09·15-s + 16-s + 2.58·17-s − 0.111·18-s + 4.58·19-s + 3.58·20-s + 5.52·21-s + 3.58·22-s − 4.27·23-s + 1.69·24-s + 7.84·25-s − 5.26·26-s − 5.28·27-s + 3.25·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.981·3-s + 0.5·4-s + 1.60·5-s + 0.693·6-s + 1.22·7-s + 0.353·8-s − 0.0370·9-s + 1.13·10-s + 1.08·11-s + 0.490·12-s − 1.45·13-s + 0.869·14-s + 1.57·15-s + 0.250·16-s + 0.627·17-s − 0.0262·18-s + 1.05·19-s + 0.801·20-s + 1.20·21-s + 0.764·22-s − 0.891·23-s + 0.346·24-s + 1.56·25-s − 1.03·26-s − 1.01·27-s + 0.614·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6014\)    =    \(2 \cdot 31 \cdot 97\)
Sign: $1$
Analytic conductor: \(48.0220\)
Root analytic conductor: \(6.92979\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6014,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.223635304\)
\(L(\frac12)\) \(\approx\) \(7.223635304\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
31 \( 1 + T \)
97 \( 1 + T \)
good3 \( 1 - 1.69T + 3T^{2} \)
5 \( 1 - 3.58T + 5T^{2} \)
7 \( 1 - 3.25T + 7T^{2} \)
11 \( 1 - 3.58T + 11T^{2} \)
13 \( 1 + 5.26T + 13T^{2} \)
17 \( 1 - 2.58T + 17T^{2} \)
19 \( 1 - 4.58T + 19T^{2} \)
23 \( 1 + 4.27T + 23T^{2} \)
29 \( 1 - 1.80T + 29T^{2} \)
37 \( 1 + 4.93T + 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 - 6.25T + 43T^{2} \)
47 \( 1 + 6.62T + 47T^{2} \)
53 \( 1 - 8.65T + 53T^{2} \)
59 \( 1 + 6.31T + 59T^{2} \)
61 \( 1 + 4.06T + 61T^{2} \)
67 \( 1 + 11.6T + 67T^{2} \)
71 \( 1 + 1.93T + 71T^{2} \)
73 \( 1 - 14.9T + 73T^{2} \)
79 \( 1 + 8.94T + 79T^{2} \)
83 \( 1 - 4.55T + 83T^{2} \)
89 \( 1 + 0.951T + 89T^{2} \)
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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

−8.014001629088321616437872991542, −7.39398454693815510440574618035, −6.61578309261559730681470135396, −5.71798716805075375391718460074, −5.24563290699149004936173314238, −4.55896711987877228039635327813, −3.51151901420479097751202727309, −2.72289449177896715109323390737, −1.94489760317468642887416744900, −1.46739879306993319503835204838, 1.46739879306993319503835204838, 1.94489760317468642887416744900, 2.72289449177896715109323390737, 3.51151901420479097751202727309, 4.55896711987877228039635327813, 5.24563290699149004936173314238, 5.71798716805075375391718460074, 6.61578309261559730681470135396, 7.39398454693815510440574618035, 8.014001629088321616437872991542

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