Properties

Degree 4
Conductor $ 11^{2} $
Sign $1$
Motivic weight 4
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s + 2·4-s + 62·5-s − 135·9-s + 22·11-s − 12·12-s − 372·15-s − 252·16-s + 124·20-s + 554·23-s + 1.63e3·25-s + 1.35e3·27-s − 2.72e3·31-s − 132·33-s − 270·36-s + 334·37-s + 44·44-s − 8.37e3·45-s + 3.40e3·47-s + 1.51e3·48-s + 1.80e3·49-s + 9.04e3·53-s + 1.36e3·55-s − 4.72e3·59-s − 744·60-s − 1.01e3·64-s − 5.60e3·67-s + ⋯
L(s)  = 1  − 2/3·3-s + 1/8·4-s + 2.47·5-s − 5/3·9-s + 2/11·11-s − 0.0833·12-s − 1.65·15-s − 0.984·16-s + 0.309·20-s + 1.04·23-s + 2.61·25-s + 1.85·27-s − 2.83·31-s − 0.121·33-s − 0.208·36-s + 0.243·37-s + 1/44·44-s − 4.13·45-s + 1.54·47-s + 0.656·48-s + 0.750·49-s + 3.21·53-s + 0.450·55-s − 1.35·59-s − 0.206·60-s − 0.248·64-s − 1.24·67-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(121\)    =    \(11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(4\)
character  :  induced by $\chi_{11} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 121,\ (\ :2, 2),\ 1)$
$L(\frac{5}{2})$  $\approx$  $1.15767$
$L(\frac12)$  $\approx$  $1.15767$
$L(3)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 11$, \(F_p\) is a polynomial of degree 4. If $p = 11$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad11$C_2$ \( 1 - 2 p T + p^{4} T^{2} \)
good2$C_2^2$ \( 1 - p T^{2} + p^{8} T^{4} \)
3$C_2$ \( ( 1 + p T + p^{4} T^{2} )^{2} \)
5$C_2$ \( ( 1 - 31 T + p^{4} T^{2} )^{2} \)
7$C_2^2$ \( 1 - 1802 T^{2} + p^{8} T^{4} \)
13$C_2^2$ \( 1 - 22442 T^{2} + p^{8} T^{4} \)
17$C_2^2$ \( 1 - 114122 T^{2} + p^{8} T^{4} \)
19$C_2^2$ \( 1 - 250922 T^{2} + p^{8} T^{4} \)
23$C_2$ \( ( 1 - 277 T + p^{4} T^{2} )^{2} \)
29$C_2$ \( ( 1 - 38 p T + p^{4} T^{2} )( 1 + 38 p T + p^{4} T^{2} ) \)
31$C_2$ \( ( 1 + 1363 T + p^{4} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 167 T + p^{4} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 4522442 T^{2} + p^{8} T^{4} \)
43$C_2^2$ \( 1 - 5385602 T^{2} + p^{8} T^{4} \)
47$C_2$ \( ( 1 - 1702 T + p^{4} T^{2} )^{2} \)
53$C_2$ \( ( 1 - 4522 T + p^{4} T^{2} )^{2} \)
59$C_2$ \( ( 1 + 2363 T + p^{4} T^{2} )^{2} \)
61$C_2^2$ \( 1 - 11966402 T^{2} + p^{8} T^{4} \)
67$C_2$ \( ( 1 + 2803 T + p^{4} T^{2} )^{2} \)
71$C_2$ \( ( 1 - 3397 T + p^{4} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 45779402 T^{2} + p^{8} T^{4} \)
79$C_2^2$ \( 1 - 40803842 T^{2} + p^{8} T^{4} \)
83$C_2^2$ \( 1 - 94223522 T^{2} + p^{8} T^{4} \)
89$C_2$ \( ( 1 + 4673 T + p^{4} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 4247 T + p^{4} T^{2} )^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−20.26132514032608998620177529834, −19.76891243336154225506294972049, −18.23883335823624907732354596626, −18.18253336283652801360653841472, −17.34444638482209044977961468755, −16.87079131996468627874928504157, −16.56304172213857674847604833284, −15.07365855119871938737353357775, −14.28695245088176938995164735449, −13.74945259582314837992793239083, −13.16983060155429795253515143690, −12.05413188563647012617656866518, −11.05392227188633231096315632129, −10.52479981427678601601763710300, −9.204238464977936325510442516002, −8.991403689752017235867616356036, −6.90260280784055901568593389407, −5.73105080156715589507556740723, −5.53102188592883621196977418424, −2.36292077307959260973677058117, 2.36292077307959260973677058117, 5.53102188592883621196977418424, 5.73105080156715589507556740723, 6.90260280784055901568593389407, 8.991403689752017235867616356036, 9.204238464977936325510442516002, 10.52479981427678601601763710300, 11.05392227188633231096315632129, 12.05413188563647012617656866518, 13.16983060155429795253515143690, 13.74945259582314837992793239083, 14.28695245088176938995164735449, 15.07365855119871938737353357775, 16.56304172213857674847604833284, 16.87079131996468627874928504157, 17.34444638482209044977961468755, 18.18253336283652801360653841472, 18.23883335823624907732354596626, 19.76891243336154225506294972049, 20.26132514032608998620177529834

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