Properties

Degree 4
Conductor $ 2^{2} \cdot 7^{4} $
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  + 2-s − 2·3-s − 2·6-s − 8-s + 3·9-s + 8·13-s − 16-s + 6·17-s + 3·18-s + 2·19-s + 2·24-s + 5·25-s + 8·26-s − 10·27-s − 12·29-s − 4·31-s + 6·34-s − 2·37-s + 2·38-s − 16·39-s − 12·41-s + 16·43-s − 12·47-s + 2·48-s + 5·50-s − 12·51-s − 6·53-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s − 0.816·6-s − 0.353·8-s + 9-s + 2.21·13-s − 1/4·16-s + 1.45·17-s + 0.707·18-s + 0.458·19-s + 0.408·24-s + 25-s + 1.56·26-s − 1.92·27-s − 2.22·29-s − 0.718·31-s + 1.02·34-s − 0.328·37-s + 0.324·38-s − 2.56·39-s − 1.87·41-s + 2.43·43-s − 1.75·47-s + 0.288·48-s + 0.707·50-s − 1.68·51-s − 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(9604\)    =    \(2^{2} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{98} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 9604,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.00395$
$L(\frac12)$  $\approx$  $1.00395$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 - T + T^{2} \)
7 \( 1 \)
good3$C_2^2$ \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−14.35056284651231564581666825333, −13.39276982253282504579114951402, −13.12545459994660673294455378565, −12.75709997600820319402207612824, −12.06926607584256368231314481450, −11.55908781045891273671686991799, −11.03156301358370933522671055749, −10.83699373690739446285393590903, −9.969986304007682103457348749805, −9.390512869525057287713939002024, −8.801779702387547062476919061217, −7.937843169684972630689982770456, −7.38187572562548851992014849471, −6.53895455765916395067666626447, −5.90847417571958222976833557124, −5.54613123540807352222510231172, −4.92836069455357757789022627749, −3.65733990102187077391429675053, −3.60730141195257487979543238621, −1.47640700840045322049156739130, 1.47640700840045322049156739130, 3.60730141195257487979543238621, 3.65733990102187077391429675053, 4.92836069455357757789022627749, 5.54613123540807352222510231172, 5.90847417571958222976833557124, 6.53895455765916395067666626447, 7.38187572562548851992014849471, 7.937843169684972630689982770456, 8.801779702387547062476919061217, 9.390512869525057287713939002024, 9.969986304007682103457348749805, 10.83699373690739446285393590903, 11.03156301358370933522671055749, 11.55908781045891273671686991799, 12.06926607584256368231314481450, 12.75709997600820319402207612824, 13.12545459994660673294455378565, 13.39276982253282504579114951402, 14.35056284651231564581666825333

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