Properties

Label 2-6014-1.1-c1-0-105
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 − 0.370·3-s + 4-s + 2.36·5-s − 0.370·6-s + 1.74·7-s + 8-s − 2.86·9-s + 2.36·10-s − 0.565·11-s − 0.370·12-s − 1.80·13-s + 1.74·14-s − 0.876·15-s + 16-s + 4.37·17-s − 2.86·18-s + 2.88·19-s + 2.36·20-s − 0.646·21-s − 0.565·22-s − 0.747·23-s − 0.370·24-s + 0.596·25-s − 1.80·26-s + 2.17·27-s + 1.74·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.213·3-s + 0.5·4-s + 1.05·5-s − 0.151·6-s + 0.660·7-s + 0.353·8-s − 0.954·9-s + 0.748·10-s − 0.170·11-s − 0.106·12-s − 0.500·13-s + 0.466·14-s − 0.226·15-s + 0.250·16-s + 1.06·17-s − 0.674·18-s + 0.661·19-s + 0.528·20-s − 0.141·21-s − 0.120·22-s − 0.155·23-s − 0.0756·24-s + 0.119·25-s − 0.353·26-s + 0.417·27-s + 0.330·28-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

\(L(1)\) \(\approx\) \(3.887722335\)
\(L(\frac12)\) \(\approx\) \(3.887722335\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.370T + 3T^{2} \)
5 \( 1 - 2.36T + 5T^{2} \)
7 \( 1 - 1.74T + 7T^{2} \)
11 \( 1 + 0.565T + 11T^{2} \)
13 \( 1 + 1.80T + 13T^{2} \)
17 \( 1 - 4.37T + 17T^{2} \)
19 \( 1 - 2.88T + 19T^{2} \)
23 \( 1 + 0.747T + 23T^{2} \)
29 \( 1 - 7.97T + 29T^{2} \)
37 \( 1 - 4.19T + 37T^{2} \)
41 \( 1 + 3.11T + 41T^{2} \)
43 \( 1 - 3.55T + 43T^{2} \)
47 \( 1 + 0.928T + 47T^{2} \)
53 \( 1 + 5.73T + 53T^{2} \)
59 \( 1 - 9.35T + 59T^{2} \)
61 \( 1 - 4.49T + 61T^{2} \)
67 \( 1 + 3.08T + 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 - 0.723T + 73T^{2} \)
79 \( 1 + 3.87T + 79T^{2} \)
83 \( 1 + 1.70T + 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
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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

−8.004219464614105438358387289228, −7.32355230600747057585923744440, −6.33255512117815800620104847958, −5.88666820147865624185436124614, −5.14864768294367464109518918109, −4.81491563210180056183217102853, −3.56306916730979067326333650756, −2.76520040827713227347210298771, −2.04994076466606593587258674280, −0.971099858912182739183554789284, 0.971099858912182739183554789284, 2.04994076466606593587258674280, 2.76520040827713227347210298771, 3.56306916730979067326333650756, 4.81491563210180056183217102853, 5.14864768294367464109518918109, 5.88666820147865624185436124614, 6.33255512117815800620104847958, 7.32355230600747057585923744440, 8.004219464614105438358387289228

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