Properties

Label 4-307328-1.1-c1e2-0-26
Degree $4$
Conductor $307328$
Sign $-1$
Analytic cond. $19.5954$
Root an. cond. $2.10396$
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-s + 4-s − 8-s + 2·9-s + 16-s − 2·18-s − 10·25-s − 32-s + 2·36-s − 16·43-s + 10·50-s + 64-s − 8·67-s − 2·72-s − 5·81-s + 16·86-s − 10·100-s + 24·107-s − 12·113-s − 22·121-s + 127-s − 128-s + 131-s + 8·134-s + 137-s + 139-s + 2·144-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s + 2/3·9-s + 1/4·16-s − 0.471·18-s − 2·25-s − 0.176·32-s + 1/3·36-s − 2.43·43-s + 1.41·50-s + 1/8·64-s − 0.977·67-s − 0.235·72-s − 5/9·81-s + 1.72·86-s − 100-s + 2.32·107-s − 1.12·113-s − 2·121-s + 0.0887·127-s − 0.0883·128-s + 0.0873·131-s + 0.691·134-s + 0.0854·137-s + 0.0848·139-s + 1/6·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 307328 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 307328 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(307328\)    =    \(2^{7} \cdot 7^{4}\)
Sign: $-1$
Analytic conductor: \(19.5954\)
Root analytic conductor: \(2.10396\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 307328,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
7 \( 1 \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−8.580252924082452784437577521921, −8.060565227517463675213125122618, −7.76701697460455810212222197120, −7.23903986714456677370051790851, −6.82198631932231330828991505682, −6.25618108569924481357234421415, −5.87678432003702945769945994384, −5.19414551646172023462303156944, −4.69603295955859913515540337928, −3.94991817797036210041836791964, −3.55057513939704101440793360919, −2.74279276132182192115698045491, −1.94943093609629836158974171979, −1.39182617144060518575568501089, 0, 1.39182617144060518575568501089, 1.94943093609629836158974171979, 2.74279276132182192115698045491, 3.55057513939704101440793360919, 3.94991817797036210041836791964, 4.69603295955859913515540337928, 5.19414551646172023462303156944, 5.87678432003702945769945994384, 6.25618108569924481357234421415, 6.82198631932231330828991505682, 7.23903986714456677370051790851, 7.76701697460455810212222197120, 8.060565227517463675213125122618, 8.580252924082452784437577521921

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