Properties

Degree 4
Conductor $ 3^{2} \cdot 5^{2} \cdot 7^{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  + 2·2-s − 3-s + 2·4-s − 5-s − 2·6-s − 7-s + 4·8-s − 2·10-s + 6·11-s − 2·12-s − 6·13-s − 2·14-s + 15-s + 8·16-s + 4·17-s − 19-s − 2·20-s + 21-s + 12·22-s + 4·23-s − 4·24-s − 12·26-s + 27-s − 2·28-s − 16·29-s + 2·30-s − 31-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s − 0.377·7-s + 1.41·8-s − 0.632·10-s + 1.80·11-s − 0.577·12-s − 1.66·13-s − 0.534·14-s + 0.258·15-s + 2·16-s + 0.970·17-s − 0.229·19-s − 0.447·20-s + 0.218·21-s + 2.55·22-s + 0.834·23-s − 0.816·24-s − 2.35·26-s + 0.192·27-s − 0.377·28-s − 2.97·29-s + 0.365·30-s − 0.179·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(11025\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{105} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 11025,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.65962$
$L(\frac12)$  $\approx$  $1.65962$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ \( 1 + T + T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 + T + p T^{2} \)
good2$C_2^2$ \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \)
67$C_2^2$ \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + T - 72 T^{2} + p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−14.36217335027837631298114730930, −13.31427756038192158740690064611, −13.06727659824459333800968084157, −12.42797169331384226126533638833, −12.09946190336053656384798194936, −11.59567067679566165484138677356, −11.18868243653864648732214163107, −10.50648944710682975799363096614, −9.764649106786253356832991372553, −9.458280534738678381554534457946, −8.540374159858362232788619976984, −7.62877944110981313382173921888, −7.13040338585399542146241657938, −6.74599157926910595495511504158, −5.81423333869221498209692112067, −5.17542032033890415550538412225, −4.77609586379963178204813101166, −3.65736222745361496227383056214, −3.60095122403794590604848515797, −1.83324032084037973742263210554, 1.83324032084037973742263210554, 3.60095122403794590604848515797, 3.65736222745361496227383056214, 4.77609586379963178204813101166, 5.17542032033890415550538412225, 5.81423333869221498209692112067, 6.74599157926910595495511504158, 7.13040338585399542146241657938, 7.62877944110981313382173921888, 8.540374159858362232788619976984, 9.458280534738678381554534457946, 9.764649106786253356832991372553, 10.50648944710682975799363096614, 11.18868243653864648732214163107, 11.59567067679566165484138677356, 12.09946190336053656384798194936, 12.42797169331384226126533638833, 13.06727659824459333800968084157, 13.31427756038192158740690064611, 14.36217335027837631298114730930

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