Properties

Degree 4
Conductor $ 2^{2} \cdot 5^{2} \cdot 29 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 5-s − 11-s + 16-s − 5·19-s − 20-s − 4·25-s − 6·29-s − 31-s + 9·41-s + 44-s + 5·49-s − 55-s + 14·61-s − 64-s − 71-s + 5·76-s − 15·79-s + 80-s − 9·81-s + 15·89-s − 5·95-s + 4·100-s + 19·101-s + 20·109-s + 6·116-s − 15·121-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.447·5-s − 0.301·11-s + 1/4·16-s − 1.14·19-s − 0.223·20-s − 4/5·25-s − 1.11·29-s − 0.179·31-s + 1.40·41-s + 0.150·44-s + 5/7·49-s − 0.134·55-s + 1.79·61-s − 1/8·64-s − 0.118·71-s + 0.573·76-s − 1.68·79-s + 0.111·80-s − 81-s + 1.58·89-s − 0.512·95-s + 2/5·100-s + 1.89·101-s + 1.91·109-s + 0.557·116-s − 1.36·121-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(2900\)    =    \(2^{2} \cdot 5^{2} \cdot 29\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{2900} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 2900,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.6834152878$
$L(\frac12)$  $\approx$  $0.6834152878$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;29\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;29\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - T + p T^{2} \)
29$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 5 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 60 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + p T^{2} ) \)
97$C_2^2$ \( 1 + 150 T^{2} + p^{2} T^{4} \)
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\[\begin{aligned} L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.97300761202007026711745745915, −12.48015169428370421085238309500, −11.58970666047779625110098362727, −11.09080844592056808048366738307, −10.31061285899280809288523356227, −9.878830696598072344068565692663, −9.107759636504157791839335091252, −8.627910123400402193922707762082, −7.80232759810327212643626620231, −7.14372402130757995244587466183, −6.10113281784815210264840570215, −5.59836080241083055223088887588, −4.57544865124138979668811884308, −3.72787597549157883263338949238, −2.25054967550044909031450405591, 2.25054967550044909031450405591, 3.72787597549157883263338949238, 4.57544865124138979668811884308, 5.59836080241083055223088887588, 6.10113281784815210264840570215, 7.14372402130757995244587466183, 7.80232759810327212643626620231, 8.627910123400402193922707762082, 9.107759636504157791839335091252, 9.878830696598072344068565692663, 10.31061285899280809288523356227, 11.09080844592056808048366738307, 11.58970666047779625110098362727, 12.48015169428370421085238309500, 12.97300761202007026711745745915

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