Properties

Label 4-1815e2-1.1-c1e2-0-36
Degree $4$
Conductor $3294225$
Sign $1$
Analytic cond. $210.042$
Root an. cond. $3.80694$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4·4-s − 2·5-s + 3·9-s + 8·12-s + 4·15-s + 12·16-s + 8·20-s + 12·23-s + 3·25-s − 4·27-s + 2·31-s − 12·36-s − 10·37-s − 6·45-s − 24·47-s − 24·48-s − 11·49-s + 12·53-s − 16·60-s − 32·64-s − 10·67-s − 24·69-s − 12·71-s − 6·75-s − 24·80-s + 5·81-s + ⋯
L(s)  = 1  − 1.15·3-s − 2·4-s − 0.894·5-s + 9-s + 2.30·12-s + 1.03·15-s + 3·16-s + 1.78·20-s + 2.50·23-s + 3/5·25-s − 0.769·27-s + 0.359·31-s − 2·36-s − 1.64·37-s − 0.894·45-s − 3.50·47-s − 3.46·48-s − 1.57·49-s + 1.64·53-s − 2.06·60-s − 4·64-s − 1.22·67-s − 2.88·69-s − 1.42·71-s − 0.692·75-s − 2.68·80-s + 5/9·81-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: induced by $\chi_{1815} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
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$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 143 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 118 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
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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

−8.967386097377301713078020815996, −8.715422139509778262017307101792, −8.312451801137936375363717875728, −8.096375974871344841906660400344, −7.30143486974116417653142659763, −7.29921933322431972523395961305, −6.48616862248567558759622718411, −6.45762518405874774990241488267, −5.47463473472307727305398745504, −5.30679273967680233329666995509, −5.04796426311487769793515776801, −4.55249069074579412970109114024, −4.30206529941477171525949680124, −3.72760061035960114855753063621, −3.23018612043223906164273475939, −2.91860150094527516812052659571, −1.34769574688992792030614946027, −1.25475203166947893171343135556, 0, 0, 1.25475203166947893171343135556, 1.34769574688992792030614946027, 2.91860150094527516812052659571, 3.23018612043223906164273475939, 3.72760061035960114855753063621, 4.30206529941477171525949680124, 4.55249069074579412970109114024, 5.04796426311487769793515776801, 5.30679273967680233329666995509, 5.47463473472307727305398745504, 6.45762518405874774990241488267, 6.48616862248567558759622718411, 7.29921933322431972523395961305, 7.30143486974116417653142659763, 8.096375974871344841906660400344, 8.312451801137936375363717875728, 8.715422139509778262017307101792, 8.967386097377301713078020815996

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