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 + 1.66·3-s + 4-s + 0.0681·5-s − 1.66·6-s + 0.103·7-s − 8-s − 0.238·9-s − 0.0681·10-s − 0.129·11-s + 1.66·12-s − 1.57·13-s − 0.103·14-s + 0.113·15-s + 16-s + 4.28·17-s + 0.238·18-s + 4.77·19-s + 0.0681·20-s + 0.171·21-s + 0.129·22-s − 23-s − 1.66·24-s − 4.99·25-s + 1.57·26-s − 5.38·27-s + 0.103·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.959·3-s + 0.5·4-s + 0.0304·5-s − 0.678·6-s + 0.0389·7-s − 0.353·8-s − 0.0796·9-s − 0.0215·10-s − 0.0389·11-s + 0.479·12-s − 0.436·13-s − 0.0275·14-s + 0.0292·15-s + 0.250·16-s + 1.03·17-s + 0.0563·18-s + 1.09·19-s + 0.0152·20-s + 0.0374·21-s + 0.0275·22-s − 0.208·23-s − 0.339·24-s − 0.999·25-s + 0.308·26-s − 1.03·27-s + 0.0194·28-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 \( 1 - 1.66T + 3T^{2} \)
5 \( 1 - 0.0681T + 5T^{2} \)
7 \( 1 - 0.103T + 7T^{2} \)
11 \( 1 + 0.129T + 11T^{2} \)
13 \( 1 + 1.57T + 13T^{2} \)
17 \( 1 - 4.28T + 17T^{2} \)
19 \( 1 - 4.77T + 19T^{2} \)
29 \( 1 + 5.98T + 29T^{2} \)
31 \( 1 - 3.37T + 31T^{2} \)
37 \( 1 + 6.98T + 37T^{2} \)
41 \( 1 + 0.948T + 41T^{2} \)
43 \( 1 + 13.0T + 43T^{2} \)
47 \( 1 + 0.273T + 47T^{2} \)
53 \( 1 + 3.18T + 53T^{2} \)
59 \( 1 + 9.41T + 59T^{2} \)
61 \( 1 + 4.95T + 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 - 3.25T + 71T^{2} \)
73 \( 1 - 2.62T + 73T^{2} \)
79 \( 1 - 2.17T + 79T^{2} \)
83 \( 1 + 2.80T + 83T^{2} \)
89 \( 1 - 8.53T + 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−7.893709670474190896422604830756, −7.36644362254054552536695305502, −6.47150922595800089345226821897, −5.60315774786590728754603043432, −4.94613332366683010549413992621, −3.55170959749929836677235566397, −3.27544487333815934643637858239, −2.22159827086186225340508766075, −1.46976239362545130181714422663, 0, 1.46976239362545130181714422663, 2.22159827086186225340508766075, 3.27544487333815934643637858239, 3.55170959749929836677235566397, 4.94613332366683010549413992621, 5.60315774786590728754603043432, 6.47150922595800089345226821897, 7.36644362254054552536695305502, 7.893709670474190896422604830756

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