Properties

Degree 2
Conductor $ 2 \cdot 23 \cdot 131 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2.94·3-s + 4-s − 3.10·5-s + 2.94·6-s + 2.18·7-s − 8-s + 5.67·9-s + 3.10·10-s + 3.62·11-s − 2.94·12-s + 2.36·13-s − 2.18·14-s + 9.13·15-s + 16-s + 2.10·17-s − 5.67·18-s − 7.83·19-s − 3.10·20-s − 6.44·21-s − 3.62·22-s + 23-s + 2.94·24-s + 4.63·25-s − 2.36·26-s − 7.86·27-s + 2.18·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.70·3-s + 0.5·4-s − 1.38·5-s + 1.20·6-s + 0.826·7-s − 0.353·8-s + 1.89·9-s + 0.981·10-s + 1.09·11-s − 0.850·12-s + 0.656·13-s − 0.584·14-s + 2.35·15-s + 0.250·16-s + 0.510·17-s − 1.33·18-s − 1.79·19-s − 0.693·20-s − 1.40·21-s − 0.773·22-s + 0.208·23-s + 0.601·24-s + 0.926·25-s − 0.464·26-s − 1.51·27-s + 0.413·28-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

\( d \)  =  \(2\)
\( N \)  =  \(6026\)    =    \(2 \cdot 23 \cdot 131\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6026} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6026,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;23,\;131\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;23,\;131\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
23 \( 1 - T \)
131 \( 1 - T \)
good3 \( 1 + 2.94T + 3T^{2} \)
5 \( 1 + 3.10T + 5T^{2} \)
7 \( 1 - 2.18T + 7T^{2} \)
11 \( 1 - 3.62T + 11T^{2} \)
13 \( 1 - 2.36T + 13T^{2} \)
17 \( 1 - 2.10T + 17T^{2} \)
19 \( 1 + 7.83T + 19T^{2} \)
29 \( 1 + 6.98T + 29T^{2} \)
31 \( 1 - 8.62T + 31T^{2} \)
37 \( 1 + 3.55T + 37T^{2} \)
41 \( 1 - 8.94T + 41T^{2} \)
43 \( 1 + 6.48T + 43T^{2} \)
47 \( 1 + 5.16T + 47T^{2} \)
53 \( 1 + 0.484T + 53T^{2} \)
59 \( 1 + 2.97T + 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 - 2.65T + 67T^{2} \)
71 \( 1 - 11.1T + 71T^{2} \)
73 \( 1 - 3.40T + 73T^{2} \)
79 \( 1 + 17.7T + 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.79995358090424120745318321620, −6.90567525270558006779631581355, −6.43323339159456121304890420821, −5.77219543188572117668307237562, −4.71621945632185351551780308373, −4.27506962481323808038374470871, −3.46685605747735302765807216707, −1.79111627000724106863302766035, −0.969732710232950058690111136428, 0, 0.969732710232950058690111136428, 1.79111627000724106863302766035, 3.46685605747735302765807216707, 4.27506962481323808038374470871, 4.71621945632185351551780308373, 5.77219543188572117668307237562, 6.43323339159456121304890420821, 6.90567525270558006779631581355, 7.79995358090424120745318321620

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