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 + 1.48·3-s + 4-s + 3.56·5-s − 1.48·6-s − 2.69·7-s − 8-s − 0.780·9-s − 3.56·10-s − 4.79·11-s + 1.48·12-s + 0.746·13-s + 2.69·14-s + 5.31·15-s + 16-s + 4.10·17-s + 0.780·18-s + 4.65·19-s + 3.56·20-s − 4.01·21-s + 4.79·22-s − 23-s − 1.48·24-s + 7.71·25-s − 0.746·26-s − 5.63·27-s − 2.69·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.860·3-s + 0.5·4-s + 1.59·5-s − 0.608·6-s − 1.01·7-s − 0.353·8-s − 0.260·9-s − 1.12·10-s − 1.44·11-s + 0.430·12-s + 0.207·13-s + 0.720·14-s + 1.37·15-s + 0.250·16-s + 0.994·17-s + 0.183·18-s + 1.06·19-s + 0.797·20-s − 0.876·21-s + 1.02·22-s − 0.208·23-s − 0.304·24-s + 1.54·25-s − 0.146·26-s − 1.08·27-s − 0.509·28-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 \( 1 - 1.48T + 3T^{2} \)
5 \( 1 - 3.56T + 5T^{2} \)
7 \( 1 + 2.69T + 7T^{2} \)
11 \( 1 + 4.79T + 11T^{2} \)
13 \( 1 - 0.746T + 13T^{2} \)
17 \( 1 - 4.10T + 17T^{2} \)
19 \( 1 - 4.65T + 19T^{2} \)
29 \( 1 + 6.16T + 29T^{2} \)
31 \( 1 + 8.26T + 31T^{2} \)
37 \( 1 + 2.87T + 37T^{2} \)
41 \( 1 + 5.19T + 41T^{2} \)
43 \( 1 - 1.25T + 43T^{2} \)
47 \( 1 + 6.72T + 47T^{2} \)
53 \( 1 - 2.81T + 53T^{2} \)
59 \( 1 - 5.55T + 59T^{2} \)
61 \( 1 + 3.81T + 61T^{2} \)
67 \( 1 + 10.3T + 67T^{2} \)
71 \( 1 - 9.37T + 71T^{2} \)
73 \( 1 + 2.73T + 73T^{2} \)
79 \( 1 + 5.86T + 79T^{2} \)
83 \( 1 + 2.35T + 83T^{2} \)
89 \( 1 + 2.14T + 89T^{2} \)
97 \( 1 + 5.78T + 97T^{2} \)
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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

−7.74594909579433598754967374594, −7.25253559071704180056084004788, −6.29140990074770562184514664137, −5.52375504798965815610609404754, −5.35214004820775784357527707965, −3.50344808802219484078498934956, −3.05448348791215272418115042995, −2.29987187469214904338629027113, −1.54416367102557312096080654637, 0, 1.54416367102557312096080654637, 2.29987187469214904338629027113, 3.05448348791215272418115042995, 3.50344808802219484078498934956, 5.35214004820775784357527707965, 5.52375504798965815610609404754, 6.29140990074770562184514664137, 7.25253559071704180056084004788, 7.74594909579433598754967374594

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