Properties

Label 4-84e4-1.1-c1e2-0-25
Degree $4$
Conductor $49787136$
Sign $1$
Analytic cond. $3174.47$
Root an. cond. $7.50616$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 14·19-s − 2·25-s − 10·31-s + 2·37-s − 12·47-s − 12·83-s − 10·103-s + 10·109-s − 12·113-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 3.21·19-s − 2/5·25-s − 1.79·31-s + 0.328·37-s − 1.75·47-s − 1.31·83-s − 0.985·103-s + 0.957·109-s − 1.12·113-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 + 0.0783·163-s + 0.0773·167-s − 0.0769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(49787136\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(3174.47\)
Root analytic conductor: \(7.50616\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 49787136,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 131 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 71 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 85 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
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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

−7.83443919419325035745512763868, −7.34472659428003179872563519637, −7.05110237479008899866380136336, −6.64080636358755861323855155350, −6.50927703924062294126202856760, −5.91714743103051310163899327776, −5.78011445482502343222104037117, −5.40055919548440975162813340406, −4.74330398784743550467077529570, −4.56519386970328119624639030610, −4.15246343598969991644703103367, −3.90249655797020908150545433496, −3.28685296949643074929053974976, −3.06902734784367876021686623000, −2.32369431935161930439800069795, −1.94002252578087871882597502326, −1.85363544482582036444098261602, −1.01458997637171047827891017328, 0, 0, 1.01458997637171047827891017328, 1.85363544482582036444098261602, 1.94002252578087871882597502326, 2.32369431935161930439800069795, 3.06902734784367876021686623000, 3.28685296949643074929053974976, 3.90249655797020908150545433496, 4.15246343598969991644703103367, 4.56519386970328119624639030610, 4.74330398784743550467077529570, 5.40055919548440975162813340406, 5.78011445482502343222104037117, 5.91714743103051310163899327776, 6.50927703924062294126202856760, 6.64080636358755861323855155350, 7.05110237479008899866380136336, 7.34472659428003179872563519637, 7.83443919419325035745512763868

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