Properties

Label 2-6017-1.1-c1-0-30
Degree $2$
Conductor $6017$
Sign $1$
Analytic cond. $48.0459$
Root an. cond. $6.93152$
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  − 2.10·2-s − 1.55·3-s + 2.43·4-s − 0.464·5-s + 3.27·6-s − 0.781·7-s − 0.926·8-s − 0.578·9-s + 0.978·10-s − 11-s − 3.79·12-s − 6.89·13-s + 1.64·14-s + 0.722·15-s − 2.92·16-s + 4.41·17-s + 1.21·18-s + 6.82·19-s − 1.13·20-s + 1.21·21-s + 2.10·22-s − 0.0938·23-s + 1.44·24-s − 4.78·25-s + 14.5·26-s + 5.56·27-s − 1.90·28-s + ⋯
L(s)  = 1  − 1.48·2-s − 0.898·3-s + 1.21·4-s − 0.207·5-s + 1.33·6-s − 0.295·7-s − 0.327·8-s − 0.192·9-s + 0.309·10-s − 0.301·11-s − 1.09·12-s − 1.91·13-s + 0.440·14-s + 0.186·15-s − 0.731·16-s + 1.07·17-s + 0.287·18-s + 1.56·19-s − 0.253·20-s + 0.265·21-s + 0.449·22-s − 0.0195·23-s + 0.294·24-s − 0.956·25-s + 2.84·26-s + 1.07·27-s − 0.360·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $1$
Analytic conductor: \(48.0459\)
Root analytic conductor: \(6.93152\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6017,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1931765716\)
\(L(\frac12)\) \(\approx\) \(0.1931765716\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + T \)
547 \( 1 - T \)
good2 \( 1 + 2.10T + 2T^{2} \)
3 \( 1 + 1.55T + 3T^{2} \)
5 \( 1 + 0.464T + 5T^{2} \)
7 \( 1 + 0.781T + 7T^{2} \)
13 \( 1 + 6.89T + 13T^{2} \)
17 \( 1 - 4.41T + 17T^{2} \)
19 \( 1 - 6.82T + 19T^{2} \)
23 \( 1 + 0.0938T + 23T^{2} \)
29 \( 1 - 6.75T + 29T^{2} \)
31 \( 1 + 9.92T + 31T^{2} \)
37 \( 1 - 1.09T + 37T^{2} \)
41 \( 1 + 10.0T + 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 + 12.6T + 47T^{2} \)
53 \( 1 - 3.80T + 53T^{2} \)
59 \( 1 + 0.976T + 59T^{2} \)
61 \( 1 + 3.58T + 61T^{2} \)
67 \( 1 + 10.4T + 67T^{2} \)
71 \( 1 + 2.15T + 71T^{2} \)
73 \( 1 - 2.99T + 73T^{2} \)
79 \( 1 - 4.36T + 79T^{2} \)
83 \( 1 + 2.51T + 83T^{2} \)
89 \( 1 - 2.79T + 89T^{2} \)
97 \( 1 + 1.81T + 97T^{2} \)
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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

−7.891775571617475677368519702224, −7.56432256108897651343660754090, −6.96960082658566039167644533487, −6.07413478674288249959382818130, −5.23369513811871983741929409031, −4.80237660831053476841172050769, −3.38408292042692404709436838387, −2.55792368017919531971202983591, −1.41637786386915938863599316697, −0.31711970818250304038173693924, 0.31711970818250304038173693924, 1.41637786386915938863599316697, 2.55792368017919531971202983591, 3.38408292042692404709436838387, 4.80237660831053476841172050769, 5.23369513811871983741929409031, 6.07413478674288249959382818130, 6.96960082658566039167644533487, 7.56432256108897651343660754090, 7.891775571617475677368519702224

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