Properties

Label 4-1815e2-1.1-c1e2-0-13
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  + 3·3-s − 3·4-s + 6·9-s − 9·12-s + 5·16-s + 25-s + 9·27-s + 4·31-s − 18·36-s + 16·37-s + 15·48-s + 5·49-s − 3·64-s + 26·67-s + 3·75-s + 9·81-s + 12·93-s − 16·97-s − 3·100-s + 16·103-s − 27·108-s + 48·111-s − 12·124-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 1.73·3-s − 3/2·4-s + 2·9-s − 2.59·12-s + 5/4·16-s + 1/5·25-s + 1.73·27-s + 0.718·31-s − 3·36-s + 2.63·37-s + 2.16·48-s + 5/7·49-s − 3/8·64-s + 3.17·67-s + 0.346·75-s + 81-s + 1.24·93-s − 1.62·97-s − 0.299·100-s + 1.57·103-s − 2.59·108-s + 4.55·111-s − 1.07·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-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

\(L(1)\) \(\approx\) \(3.398567471\)
\(L(\frac12)\) \(\approx\) \(3.398567471\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11 \( 1 \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 61 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
61$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
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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

−7.60688241377169986873087565661, −7.40090582004259536909041087857, −6.73417120558738514878326827502, −6.37439081717833847319244623854, −5.77040581787511510488276683680, −5.27624100818914759567282986235, −4.78857227701726477469182496681, −4.34439366417551788094185895121, −4.09725930317653310130576991743, −3.61306702003378233717080913968, −3.17595939682020412974726176068, −2.49706731435942794752325053650, −2.27410472873672432901979708453, −1.26432348658476793506551703218, −0.70179579221735971248477045061, 0.70179579221735971248477045061, 1.26432348658476793506551703218, 2.27410472873672432901979708453, 2.49706731435942794752325053650, 3.17595939682020412974726176068, 3.61306702003378233717080913968, 4.09725930317653310130576991743, 4.34439366417551788094185895121, 4.78857227701726477469182496681, 5.27624100818914759567282986235, 5.77040581787511510488276683680, 6.37439081717833847319244623854, 6.73417120558738514878326827502, 7.40090582004259536909041087857, 7.60688241377169986873087565661

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