Properties

Label 2-6026-1.1-c1-0-178
Degree $2$
Conductor $6026$
Sign $-1$
Analytic cond. $48.1178$
Root an. cond. $6.93670$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 0.178·3-s + 4-s − 1.41·5-s − 0.178·6-s − 1.81·7-s + 8-s − 2.96·9-s − 1.41·10-s + 2.27·11-s − 0.178·12-s + 4.66·13-s − 1.81·14-s + 0.251·15-s + 16-s − 3.31·17-s − 2.96·18-s + 0.914·19-s − 1.41·20-s + 0.323·21-s + 2.27·22-s + 23-s − 0.178·24-s − 3.00·25-s + 4.66·26-s + 1.06·27-s − 1.81·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.102·3-s + 0.5·4-s − 0.631·5-s − 0.0727·6-s − 0.686·7-s + 0.353·8-s − 0.989·9-s − 0.446·10-s + 0.685·11-s − 0.0514·12-s + 1.29·13-s − 0.485·14-s + 0.0649·15-s + 0.250·16-s − 0.804·17-s − 0.699·18-s + 0.209·19-s − 0.315·20-s + 0.0705·21-s + 0.484·22-s + 0.208·23-s − 0.0363·24-s − 0.600·25-s + 0.914·26-s + 0.204·27-s − 0.343·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6026\)    =    \(2 \cdot 23 \cdot 131\)
Sign: $-1$
Analytic conductor: \(48.1178\)
Root analytic conductor: \(6.93670\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6026,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
23 \( 1 - T \)
131 \( 1 + T \)
good3 \( 1 + 0.178T + 3T^{2} \)
5 \( 1 + 1.41T + 5T^{2} \)
7 \( 1 + 1.81T + 7T^{2} \)
11 \( 1 - 2.27T + 11T^{2} \)
13 \( 1 - 4.66T + 13T^{2} \)
17 \( 1 + 3.31T + 17T^{2} \)
19 \( 1 - 0.914T + 19T^{2} \)
29 \( 1 + 1.00T + 29T^{2} \)
31 \( 1 - 3.63T + 31T^{2} \)
37 \( 1 - 5.78T + 37T^{2} \)
41 \( 1 + 11.9T + 41T^{2} \)
43 \( 1 + 2.18T + 43T^{2} \)
47 \( 1 - 3.07T + 47T^{2} \)
53 \( 1 - 4.22T + 53T^{2} \)
59 \( 1 + 7.70T + 59T^{2} \)
61 \( 1 - 8.68T + 61T^{2} \)
67 \( 1 + 14.8T + 67T^{2} \)
71 \( 1 - 2.58T + 71T^{2} \)
73 \( 1 + 3.33T + 73T^{2} \)
79 \( 1 + 11.6T + 79T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 - 12.9T + 89T^{2} \)
97 \( 1 - 8.79T + 97T^{2} \)
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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

−7.65964654922087753117093616132, −6.74196007392613620104679597627, −6.27624852912374717470847107226, −5.70338125743672105311606469713, −4.73816076186414625954548310772, −3.91945598185355368141863100706, −3.40834721294422073971780641129, −2.62315562177100957142623282988, −1.36950521045049169998370783175, 0, 1.36950521045049169998370783175, 2.62315562177100957142623282988, 3.40834721294422073971780641129, 3.91945598185355368141863100706, 4.73816076186414625954548310772, 5.70338125743672105311606469713, 6.27624852912374717470847107226, 6.74196007392613620104679597627, 7.65964654922087753117093616132

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