Properties

Degree 4
Conductor $ 13^{2} \cdot 31^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s + 4·4-s + 6·9-s + 16·12-s − 4·13-s + 12·16-s − 4·17-s − 8·23-s − 6·25-s − 4·27-s + 16·29-s + 24·36-s − 16·39-s + 8·43-s + 48·48-s + 10·49-s − 16·51-s − 16·52-s − 8·53-s + 20·61-s + 32·64-s − 16·68-s − 32·69-s − 24·75-s − 32·79-s − 37·81-s + 64·87-s + ⋯
L(s)  = 1  + 2.30·3-s + 2·4-s + 2·9-s + 4.61·12-s − 1.10·13-s + 3·16-s − 0.970·17-s − 1.66·23-s − 6/5·25-s − 0.769·27-s + 2.97·29-s + 4·36-s − 2.56·39-s + 1.21·43-s + 6.92·48-s + 10/7·49-s − 2.24·51-s − 2.21·52-s − 1.09·53-s + 2.56·61-s + 4·64-s − 1.94·68-s − 3.85·69-s − 2.77·75-s − 3.60·79-s − 4.11·81-s + 6.86·87-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 162409 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 162409 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(162409\)    =    \(13^{2} \cdot 31^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{403} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 162409,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $5.18628$
$L(\frac12)$  $\approx$  $5.18628$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{13,\;31\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{13,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad13$C_2$ \( 1 + 4 T + p T^{2} \)
31$C_2$ \( 1 + T^{2} \)
good2$C_2$ \( ( 1 - p T^{2} )^{2} \)
3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 47 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−11.48824188220044485522628631094, −11.11640169391882038432642624454, −10.40650301407362823743526944467, −9.950788799957586492306938655011, −9.865101921932921833798347143972, −9.148093643779623950381364834693, −8.424697193241376930589622367273, −8.330891792706934249857488501637, −7.87332931386543365109413609431, −7.39411504380701027906515046948, −7.03246694791877821157741815824, −6.42619710812602113355812489125, −5.93747167203652307881585637367, −5.34892375760208459795988448951, −4.15389208039854490017879517281, −3.91813201613630258886152375650, −2.87913545534096285272776893375, −2.67639293080402667122026786589, −2.33898214138781229335548978487, −1.67317628191153079059259795335, 1.67317628191153079059259795335, 2.33898214138781229335548978487, 2.67639293080402667122026786589, 2.87913545534096285272776893375, 3.91813201613630258886152375650, 4.15389208039854490017879517281, 5.34892375760208459795988448951, 5.93747167203652307881585637367, 6.42619710812602113355812489125, 7.03246694791877821157741815824, 7.39411504380701027906515046948, 7.87332931386543365109413609431, 8.330891792706934249857488501637, 8.424697193241376930589622367273, 9.148093643779623950381364834693, 9.865101921932921833798347143972, 9.950788799957586492306938655011, 10.40650301407362823743526944467, 11.11640169391882038432642624454, 11.48824188220044485522628631094

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