Properties

Degree 2
Conductor $ 2 \cdot 13 \cdot 1171 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 3·5-s + 6-s − 5·7-s − 8-s − 2·9-s + 3·10-s − 3·11-s − 12-s − 13-s + 5·14-s + 3·15-s + 16-s − 6·17-s + 2·18-s − 8·19-s − 3·20-s + 5·21-s + 3·22-s − 2·23-s + 24-s + 4·25-s + 26-s + 5·27-s − 5·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 1.88·7-s − 0.353·8-s − 2/3·9-s + 0.948·10-s − 0.904·11-s − 0.288·12-s − 0.277·13-s + 1.33·14-s + 0.774·15-s + 1/4·16-s − 1.45·17-s + 0.471·18-s − 1.83·19-s − 0.670·20-s + 1.09·21-s + 0.639·22-s − 0.417·23-s + 0.204·24-s + 4/5·25-s + 0.196·26-s + 0.962·27-s − 0.944·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(30446\)    =    \(2 \cdot 13 \cdot 1171\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{30446} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(2,\ 30446,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;13,\;1171\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;13,\;1171\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
13 \( 1 + T \)
1171 \( 1 + T \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 5 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−16.06583408036765, −15.49347438317300, −15.09145906067660, −14.69550699110784, −13.47551352948945, −13.13945501960538, −12.67451433998402, −12.09153916498776, −11.62793404017024, −11.02493300536577, −10.54245212070515, −10.21598964552628, −9.355116681725514, −8.779106387375761, −8.471957259128726, −7.719241017662244, −7.056939534263258, −6.644184614445545, −6.141811843238635, −5.475096579132875, −4.604178422010488, −3.910809682448113, −3.282367869967265, −2.681011658573923, −1.887061815896390, 0, 0, 0, 1.887061815896390, 2.681011658573923, 3.282367869967265, 3.910809682448113, 4.604178422010488, 5.475096579132875, 6.141811843238635, 6.644184614445545, 7.056939534263258, 7.719241017662244, 8.471957259128726, 8.779106387375761, 9.355116681725514, 10.21598964552628, 10.54245212070515, 11.02493300536577, 11.62793404017024, 12.09153916498776, 12.67451433998402, 13.13945501960538, 13.47551352948945, 14.69550699110784, 15.09145906067660, 15.49347438317300, 16.06583408036765

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