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Properties

Label 2-6026-1.1-c1-0-175
Degree $2$
Conductor $6026$
Sign $-1$
Analytic cond. $48.1178$
Root an. cond. $6.93670$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 3·5-s − 2·7-s − 8-s − 3·9-s − 3·10-s − 11-s + 4·13-s + 2·14-s + 16-s + 3·17-s + 3·18-s − 8·19-s + 3·20-s + 22-s − 23-s + 4·25-s − 4·26-s − 2·28-s + 6·29-s − 4·31-s − 32-s − 3·34-s − 6·35-s − 3·36-s + 2·37-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.755·7-s − 0.353·8-s − 9-s − 0.948·10-s − 0.301·11-s + 1.10·13-s + 0.534·14-s + 1/4·16-s + 0.727·17-s + 0.707·18-s − 1.83·19-s + 0.670·20-s + 0.213·22-s − 0.208·23-s + 4/5·25-s − 0.784·26-s − 0.377·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.514·34-s − 1.01·35-s − 1/2·36-s + 0.328·37-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: yes
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 \( 1 + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−8.036281640392138892825156813348, −6.79980994327101768881605859677, −6.22453022250084207423146146166, −5.94322122683812631838905671657, −5.10470458663840832013992512072, −3.84737624252745984010139335545, −2.93409352946649763864285112985, −2.28209819062415043664115546009, −1.32570672166437623701380861610, 0, 1.32570672166437623701380861610, 2.28209819062415043664115546009, 2.93409352946649763864285112985, 3.84737624252745984010139335545, 5.10470458663840832013992512072, 5.94322122683812631838905671657, 6.22453022250084207423146146166, 6.79980994327101768881605859677, 8.036281640392138892825156813348

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