Properties

Label 2-6046-1.1-c1-0-250
Degree $2$
Conductor $6046$
Sign $1$
Analytic cond. $48.2775$
Root an. cond. $6.94820$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 0.381·3-s + 4-s − 3·5-s − 0.381·6-s − 3.61·7-s + 8-s − 2.85·9-s − 3·10-s − 0.381·11-s − 0.381·12-s − 6.23·13-s − 3.61·14-s + 1.14·15-s + 16-s − 5·17-s − 2.85·18-s − 6.61·19-s − 3·20-s + 1.38·21-s − 0.381·22-s − 6.23·23-s − 0.381·24-s + 4·25-s − 6.23·26-s + 2.23·27-s − 3.61·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.220·3-s + 0.5·4-s − 1.34·5-s − 0.155·6-s − 1.36·7-s + 0.353·8-s − 0.951·9-s − 0.948·10-s − 0.115·11-s − 0.110·12-s − 1.72·13-s − 0.966·14-s + 0.295·15-s + 0.250·16-s − 1.21·17-s − 0.672·18-s − 1.51·19-s − 0.670·20-s + 0.301·21-s − 0.0814·22-s − 1.30·23-s − 0.0779·24-s + 0.800·25-s − 1.22·26-s + 0.430·27-s − 0.683·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6046\)    =    \(2 \cdot 3023\)
Sign: $1$
Analytic conductor: \(48.2775\)
Root analytic conductor: \(6.94820\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 6046,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3023 \( 1+O(T) \)
good3 \( 1 + 0.381T + 3T^{2} \)
5 \( 1 + 3T + 5T^{2} \)
7 \( 1 + 3.61T + 7T^{2} \)
11 \( 1 + 0.381T + 11T^{2} \)
13 \( 1 + 6.23T + 13T^{2} \)
17 \( 1 + 5T + 17T^{2} \)
19 \( 1 + 6.61T + 19T^{2} \)
23 \( 1 + 6.23T + 23T^{2} \)
29 \( 1 - 3.23T + 29T^{2} \)
31 \( 1 - 4.47T + 31T^{2} \)
37 \( 1 + 7.47T + 37T^{2} \)
41 \( 1 + 0.909T + 41T^{2} \)
43 \( 1 - 4.85T + 43T^{2} \)
47 \( 1 - 1.47T + 47T^{2} \)
53 \( 1 + 0.618T + 53T^{2} \)
59 \( 1 - 2.70T + 59T^{2} \)
61 \( 1 - 2.09T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + 3.70T + 71T^{2} \)
73 \( 1 - 8.94T + 73T^{2} \)
79 \( 1 - 0.527T + 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 - 4.47T + 89T^{2} \)
97 \( 1 + 6.47T + 97T^{2} \)
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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

−7.14986406050398670056444818843, −6.61815696621785750282684977632, −6.05030129321244386549756197301, −5.06646470461001858014284908912, −4.32846728345785024807493335882, −3.80269956502424158457407770484, −2.80207683302368755476818249028, −2.35127339045934623827273323973, 0, 0, 2.35127339045934623827273323973, 2.80207683302368755476818249028, 3.80269956502424158457407770484, 4.32846728345785024807493335882, 5.06646470461001858014284908912, 6.05030129321244386549756197301, 6.61815696621785750282684977632, 7.14986406050398670056444818843

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