Properties

Label 2-6014-1.1-c1-0-177
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 + 2.75·3-s + 4-s + 1.98·5-s + 2.75·6-s − 0.480·7-s + 8-s + 4.56·9-s + 1.98·10-s − 1.78·11-s + 2.75·12-s + 2.39·13-s − 0.480·14-s + 5.44·15-s + 16-s + 6.60·17-s + 4.56·18-s + 7.15·19-s + 1.98·20-s − 1.32·21-s − 1.78·22-s − 6.40·23-s + 2.75·24-s − 1.07·25-s + 2.39·26-s + 4.30·27-s − 0.480·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.58·3-s + 0.5·4-s + 0.885·5-s + 1.12·6-s − 0.181·7-s + 0.353·8-s + 1.52·9-s + 0.626·10-s − 0.537·11-s + 0.793·12-s + 0.664·13-s − 0.128·14-s + 1.40·15-s + 0.250·16-s + 1.60·17-s + 1.07·18-s + 1.64·19-s + 0.442·20-s − 0.288·21-s − 0.379·22-s − 1.33·23-s + 0.561·24-s − 0.215·25-s + 0.469·26-s + 0.828·27-s − 0.0908·28-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

\(L(1)\) \(\approx\) \(7.243453844\)
\(L(\frac12)\) \(\approx\) \(7.243453844\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.75T + 3T^{2} \)
5 \( 1 - 1.98T + 5T^{2} \)
7 \( 1 + 0.480T + 7T^{2} \)
11 \( 1 + 1.78T + 11T^{2} \)
13 \( 1 - 2.39T + 13T^{2} \)
17 \( 1 - 6.60T + 17T^{2} \)
19 \( 1 - 7.15T + 19T^{2} \)
23 \( 1 + 6.40T + 23T^{2} \)
29 \( 1 - 0.284T + 29T^{2} \)
37 \( 1 + 4.40T + 37T^{2} \)
41 \( 1 - 7.29T + 41T^{2} \)
43 \( 1 + 9.19T + 43T^{2} \)
47 \( 1 - 2.14T + 47T^{2} \)
53 \( 1 + 5.25T + 53T^{2} \)
59 \( 1 - 4.37T + 59T^{2} \)
61 \( 1 - 5.08T + 61T^{2} \)
67 \( 1 + 15.2T + 67T^{2} \)
71 \( 1 - 2.01T + 71T^{2} \)
73 \( 1 + 7.45T + 73T^{2} \)
79 \( 1 - 7.34T + 79T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 + 16.0T + 89T^{2} \)
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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

−7.82606478690998055628429066679, −7.69798599042885911405628640823, −6.65344355313862201102533696629, −5.73321957453332792098827199842, −5.36836052555906898899245778506, −4.20412740793034566258952223684, −3.33987001534297670638726553553, −3.06241302702323002112179136497, −2.04671534234568349359604390844, −1.36284337077564953267689590703, 1.36284337077564953267689590703, 2.04671534234568349359604390844, 3.06241302702323002112179136497, 3.33987001534297670638726553553, 4.20412740793034566258952223684, 5.36836052555906898899245778506, 5.73321957453332792098827199842, 6.65344355313862201102533696629, 7.69798599042885911405628640823, 7.82606478690998055628429066679

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