Properties

Label 4-1815e2-1.1-c1e2-0-23
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4-s + 3·9-s + 2·12-s − 3·16-s + 25-s − 4·27-s + 8·31-s − 3·36-s + 8·37-s + 12·47-s + 6·48-s − 2·49-s − 12·59-s + 7·64-s + 8·67-s − 12·71-s − 2·75-s + 5·81-s − 16·93-s + 4·97-s − 100-s + 4·103-s + 4·108-s − 16·111-s − 12·113-s − 8·124-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s + 9-s + 0.577·12-s − 3/4·16-s + 1/5·25-s − 0.769·27-s + 1.43·31-s − 1/2·36-s + 1.31·37-s + 1.75·47-s + 0.866·48-s − 2/7·49-s − 1.56·59-s + 7/8·64-s + 0.977·67-s − 1.42·71-s − 0.230·75-s + 5/9·81-s − 1.65·93-s + 0.406·97-s − 0.0999·100-s + 0.394·103-s + 0.384·108-s − 1.51·111-s − 1.12·113-s − 0.718·124-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: \(1\)
Selberg data: \((4,\ 3294225,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11 \( 1 \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−7.26865453813587007648236585531, −6.85148596046844826148453491842, −6.30895003848223476906417348596, −6.10393323956121282521751877576, −5.73064095886285491692068937765, −5.09249574448959305810139200699, −4.77801578706496934630419848355, −4.44994679873204421798537317144, −4.02803965188681937843711153412, −3.48353662209482708119770478632, −2.68072355101008303033232110220, −2.35824350904084673016574405264, −1.40326365182592763485746000445, −0.848684206917762978894320143348, 0, 0.848684206917762978894320143348, 1.40326365182592763485746000445, 2.35824350904084673016574405264, 2.68072355101008303033232110220, 3.48353662209482708119770478632, 4.02803965188681937843711153412, 4.44994679873204421798537317144, 4.77801578706496934630419848355, 5.09249574448959305810139200699, 5.73064095886285491692068937765, 6.10393323956121282521751877576, 6.30895003848223476906417348596, 6.85148596046844826148453491842, 7.26865453813587007648236585531

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