Properties

Label 4-1815e2-1.1-c1e2-0-8
Degree $4$
Conductor $3294225$
Sign $1$
Analytic cond. $210.042$
Root an. cond. $3.80694$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 3·9-s − 4·16-s − 5·25-s + 4·27-s + 10·31-s + 10·37-s − 14·47-s − 8·48-s − 9·49-s + 4·53-s − 6·59-s − 2·67-s + 6·71-s − 10·75-s + 5·81-s + 2·89-s + 20·93-s + 26·97-s + 26·103-s + 20·111-s − 14·113-s + 127-s + 131-s + 137-s + 139-s − 28·141-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s − 16-s − 25-s + 0.769·27-s + 1.79·31-s + 1.64·37-s − 2.04·47-s − 1.15·48-s − 9/7·49-s + 0.549·53-s − 0.781·59-s − 0.244·67-s + 0.712·71-s − 1.15·75-s + 5/9·81-s + 0.211·89-s + 2.07·93-s + 2.63·97-s + 2.56·103-s + 1.89·111-s − 1.31·113-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.35·141-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3294225\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(210.042\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3294225} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3294225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.308633975\)
\(L(\frac12)\) \(\approx\) \(3.308633975\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 - T )^{2} \)
5$C_2$ \( 1 + p T^{2} \)
11 \( 1 \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
7$C_2^2$ \( 1 + 9 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 3 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 3 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 107 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 115 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 47 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 134 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 - 9 T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.68794062247014462758087552021, −7.19742608968454064251358537662, −6.61289776364040204484804624675, −6.37250549919330927863485425232, −6.07004707392929795306499018708, −5.31495295155884607050472383115, −4.77290224364744032813673970877, −4.49640278582798122937392998994, −4.12324282443417323411558947963, −3.39380130341969957297479313713, −3.15317096274986611064477544023, −2.52114174269983034095386834162, −2.09484306499358959131555990430, −1.53825627706854947467848336561, −0.62330363422154110492253201967, 0.62330363422154110492253201967, 1.53825627706854947467848336561, 2.09484306499358959131555990430, 2.52114174269983034095386834162, 3.15317096274986611064477544023, 3.39380130341969957297479313713, 4.12324282443417323411558947963, 4.49640278582798122937392998994, 4.77290224364744032813673970877, 5.31495295155884607050472383115, 6.07004707392929795306499018708, 6.37250549919330927863485425232, 6.61289776364040204484804624675, 7.19742608968454064251358537662, 7.68794062247014462758087552021

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