Properties

Label 4-1815e2-1.1-c1e2-0-16
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 + 3·4-s − 2·5-s + 3·9-s + 6·12-s − 4·15-s + 5·16-s − 6·20-s − 4·23-s + 3·25-s + 4·27-s + 4·31-s + 9·36-s − 6·45-s + 4·47-s + 10·48-s + 2·49-s + 12·59-s − 12·60-s + 3·64-s − 8·69-s + 16·71-s + 6·75-s − 10·80-s + 5·81-s − 16·89-s − 12·92-s + ⋯
L(s)  = 1  + 1.15·3-s + 3/2·4-s − 0.894·5-s + 9-s + 1.73·12-s − 1.03·15-s + 5/4·16-s − 1.34·20-s − 0.834·23-s + 3/5·25-s + 0.769·27-s + 0.718·31-s + 3/2·36-s − 0.894·45-s + 0.583·47-s + 1.44·48-s + 2/7·49-s + 1.56·59-s − 1.54·60-s + 3/8·64-s − 0.963·69-s + 1.89·71-s + 0.692·75-s − 1.11·80-s + 5/9·81-s − 1.69·89-s − 1.25·92-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

\(L(1)\) \(\approx\) \(4.882011752\)
\(L(\frac12)\) \(\approx\) \(4.882011752\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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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

−7.60339523399861478911591257908, −7.10745461134828532312351558431, −6.83320522535778225263373405839, −6.45998887159628483454301412399, −6.02765959791722069150219056110, −5.40264046224584189961588499389, −4.99830573435640446190598710763, −4.20701248329351737922407289500, −4.01188876125749509547739913232, −3.52159812154109086894980433844, −2.95272499829946031906496696151, −2.59737853010778776210630568898, −2.13577887277830332901180521070, −1.57129750946872648039280326166, −0.74787223880318507666687220102, 0.74787223880318507666687220102, 1.57129750946872648039280326166, 2.13577887277830332901180521070, 2.59737853010778776210630568898, 2.95272499829946031906496696151, 3.52159812154109086894980433844, 4.01188876125749509547739913232, 4.20701248329351737922407289500, 4.99830573435640446190598710763, 5.40264046224584189961588499389, 6.02765959791722069150219056110, 6.45998887159628483454301412399, 6.83320522535778225263373405839, 7.10745461134828532312351558431, 7.60339523399861478911591257908

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