Properties

Label 2-6014-1.1-c1-0-143
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 − 0.848·3-s + 4-s + 4.13·5-s + 0.848·6-s + 1.58·7-s − 8-s − 2.28·9-s − 4.13·10-s + 5.98·11-s − 0.848·12-s + 0.0719·13-s − 1.58·14-s − 3.50·15-s + 16-s + 4.53·17-s + 2.28·18-s + 5.29·19-s + 4.13·20-s − 1.34·21-s − 5.98·22-s + 2.66·23-s + 0.848·24-s + 12.0·25-s − 0.0719·26-s + 4.47·27-s + 1.58·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.489·3-s + 0.5·4-s + 1.84·5-s + 0.346·6-s + 0.599·7-s − 0.353·8-s − 0.760·9-s − 1.30·10-s + 1.80·11-s − 0.244·12-s + 0.0199·13-s − 0.423·14-s − 0.905·15-s + 0.250·16-s + 1.10·17-s + 0.537·18-s + 1.21·19-s + 0.924·20-s − 0.293·21-s − 1.27·22-s + 0.554·23-s + 0.173·24-s + 2.41·25-s − 0.0141·26-s + 0.861·27-s + 0.299·28-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

\(L(1)\) \(\approx\) \(2.419042226\)
\(L(\frac12)\) \(\approx\) \(2.419042226\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.848T + 3T^{2} \)
5 \( 1 - 4.13T + 5T^{2} \)
7 \( 1 - 1.58T + 7T^{2} \)
11 \( 1 - 5.98T + 11T^{2} \)
13 \( 1 - 0.0719T + 13T^{2} \)
17 \( 1 - 4.53T + 17T^{2} \)
19 \( 1 - 5.29T + 19T^{2} \)
23 \( 1 - 2.66T + 23T^{2} \)
29 \( 1 - 7.48T + 29T^{2} \)
37 \( 1 + 5.22T + 37T^{2} \)
41 \( 1 - 11.5T + 41T^{2} \)
43 \( 1 + 6.07T + 43T^{2} \)
47 \( 1 - 0.00414T + 47T^{2} \)
53 \( 1 + 0.527T + 53T^{2} \)
59 \( 1 + 8.17T + 59T^{2} \)
61 \( 1 + 2.13T + 61T^{2} \)
67 \( 1 + 8.75T + 67T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + 13.8T + 73T^{2} \)
79 \( 1 - 13.0T + 79T^{2} \)
83 \( 1 + 4.79T + 83T^{2} \)
89 \( 1 + 6.54T + 89T^{2} \)
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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

−8.257743717051043327157392257219, −7.27545909323446500037191640567, −6.49461138031467567350709066451, −6.07728935368278792503931369281, −5.40786513356803797745061943703, −4.78285294659031939588563720124, −3.36371092470164203294357069412, −2.60041818720089207003016992365, −1.37914273276224709743022514831, −1.16860418762944799044130167872, 1.16860418762944799044130167872, 1.37914273276224709743022514831, 2.60041818720089207003016992365, 3.36371092470164203294357069412, 4.78285294659031939588563720124, 5.40786513356803797745061943703, 6.07728935368278792503931369281, 6.49461138031467567350709066451, 7.27545909323446500037191640567, 8.257743717051043327157392257219

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