Properties

Label 4-112e3-1.1-c1e2-0-38
Degree $4$
Conductor $1404928$
Sign $-1$
Analytic cond. $89.5794$
Root an. cond. $3.07646$
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  + 7-s + 4·9-s + 2·11-s − 2·23-s − 2·25-s − 6·29-s − 14·37-s − 6·43-s + 49-s − 8·53-s + 4·63-s − 4·67-s − 4·71-s + 2·77-s + 7·81-s + 8·99-s − 20·107-s + 2·109-s + 16·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  + 0.377·7-s + 4/3·9-s + 0.603·11-s − 0.417·23-s − 2/5·25-s − 1.11·29-s − 2.30·37-s − 0.914·43-s + 1/7·49-s − 1.09·53-s + 0.503·63-s − 0.488·67-s − 0.474·71-s + 0.227·77-s + 7/9·81-s + 0.804·99-s − 1.93·107-s + 0.191·109-s + 1.50·113-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1404928\)    =    \(2^{12} \cdot 7^{3}\)
Sign: $-1$
Analytic conductor: \(89.5794\)
Root analytic conductor: \(3.07646\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1404928,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad2 \( 1 \)
7$C_1$ \( 1 - T \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 106 T^{2} + p^{2} T^{4} \)
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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

−7.69482405895834397367412039494, −7.30643370909968080566606447899, −6.82759137091116655199567209076, −6.59719074620483121754814490161, −6.00609131104675387706915268936, −5.45987015954679960643740917137, −5.06574236903714323014596414504, −4.54542721719265047362564008627, −4.08256113897269383538103799877, −3.65194203381077251607794215944, −3.21241631286068701200752599073, −2.27372078903964464441170712081, −1.66066384870243542860380705788, −1.36585965362579962812780371947, 0, 1.36585965362579962812780371947, 1.66066384870243542860380705788, 2.27372078903964464441170712081, 3.21241631286068701200752599073, 3.65194203381077251607794215944, 4.08256113897269383538103799877, 4.54542721719265047362564008627, 5.06574236903714323014596414504, 5.45987015954679960643740917137, 6.00609131104675387706915268936, 6.59719074620483121754814490161, 6.82759137091116655199567209076, 7.30643370909968080566606447899, 7.69482405895834397367412039494

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