Properties

Degree $2$
Conductor $6039$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.61·2-s + 0.597·4-s − 4.27·5-s + 1.98·7-s + 2.25·8-s + 6.89·10-s − 11-s + 0.250·13-s − 3.20·14-s − 4.83·16-s − 3.17·17-s − 3.76·19-s − 2.55·20-s + 1.61·22-s − 8.19·23-s + 13.3·25-s − 0.404·26-s + 1.18·28-s + 4.02·29-s + 2.98·31-s + 3.27·32-s + 5.11·34-s − 8.50·35-s − 8.39·37-s + 6.06·38-s − 9.67·40-s + 2.30·41-s + ⋯
L(s)  = 1  − 1.13·2-s + 0.298·4-s − 1.91·5-s + 0.751·7-s + 0.798·8-s + 2.18·10-s − 0.301·11-s + 0.0695·13-s − 0.856·14-s − 1.20·16-s − 0.769·17-s − 0.863·19-s − 0.572·20-s + 0.343·22-s − 1.70·23-s + 2.66·25-s − 0.0792·26-s + 0.224·28-s + 0.746·29-s + 0.535·31-s + 0.579·32-s + 0.876·34-s − 1.43·35-s − 1.37·37-s + 0.984·38-s − 1.52·40-s + 0.360·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6039\)    =    \(3^{2} \cdot 11 \cdot 61\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{6039} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6039,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2688801596\)
\(L(\frac12)\) \(\approx\) \(0.2688801596\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + T \)
61 \( 1 + T \)
good2 \( 1 + 1.61T + 2T^{2} \)
5 \( 1 + 4.27T + 5T^{2} \)
7 \( 1 - 1.98T + 7T^{2} \)
13 \( 1 - 0.250T + 13T^{2} \)
17 \( 1 + 3.17T + 17T^{2} \)
19 \( 1 + 3.76T + 19T^{2} \)
23 \( 1 + 8.19T + 23T^{2} \)
29 \( 1 - 4.02T + 29T^{2} \)
31 \( 1 - 2.98T + 31T^{2} \)
37 \( 1 + 8.39T + 37T^{2} \)
41 \( 1 - 2.30T + 41T^{2} \)
43 \( 1 - 7.83T + 43T^{2} \)
47 \( 1 + 0.852T + 47T^{2} \)
53 \( 1 - 7.07T + 53T^{2} \)
59 \( 1 + 9.77T + 59T^{2} \)
67 \( 1 + 13.1T + 67T^{2} \)
71 \( 1 + 2.07T + 71T^{2} \)
73 \( 1 + 1.20T + 73T^{2} \)
79 \( 1 - 2.92T + 79T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 - 9.78T + 89T^{2} \)
97 \( 1 - 3.01T + 97T^{2} \)
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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

−8.282038304282106674825795691274, −7.60562678618518017653469665656, −7.11475246551731068993630947941, −6.19469366272426850055782536785, −4.88311618937908886422861028239, −4.37131478925420016220633557916, −3.86035405469692627298086103850, −2.61610666110523402284048799793, −1.52743929085332522593763669464, −0.33480758345315629334503361709, 0.33480758345315629334503361709, 1.52743929085332522593763669464, 2.61610666110523402284048799793, 3.86035405469692627298086103850, 4.37131478925420016220633557916, 4.88311618937908886422861028239, 6.19469366272426850055782536785, 7.11475246551731068993630947941, 7.60562678618518017653469665656, 8.282038304282106674825795691274

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