Properties

Label 4-176e2-1.1-c1e2-0-4
Degree $4$
Conductor $30976$
Sign $-1$
Analytic cond. $1.97505$
Root an. cond. $1.18548$
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  − 2·5-s − 5·9-s − 7·25-s − 10·37-s + 10·45-s + 2·49-s − 20·53-s + 16·81-s − 2·89-s − 2·97-s + 38·113-s − 11·121-s + 26·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 20·185-s + ⋯
L(s)  = 1  − 0.894·5-s − 5/3·9-s − 7/5·25-s − 1.64·37-s + 1.49·45-s + 2/7·49-s − 2.74·53-s + 16/9·81-s − 0.211·89-s − 0.203·97-s + 3.57·113-s − 121-s + 2.32·125-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 + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 1.47·185-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(30976\)    =    \(2^{8} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(1.97505\)
Root analytic conductor: \(1.18548\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 30976,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
11$C_2$ \( 1 + p T^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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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

−10.27471088867188668939987276364, −9.659062125287674825647078768492, −9.100419443419261215797783365286, −8.549240722330138415006197020084, −8.122888926522454399718102217182, −7.69283919060283494613682724389, −7.04739482916896435755820863629, −6.26258704707356015877913199355, −5.80278723044902588821713250020, −5.15450424761623926257405798721, −4.42307590225745849994228488667, −3.55280321483697006393530382612, −3.11624791068925415362509305436, −2.00999291903068182987488885475, 0, 2.00999291903068182987488885475, 3.11624791068925415362509305436, 3.55280321483697006393530382612, 4.42307590225745849994228488667, 5.15450424761623926257405798721, 5.80278723044902588821713250020, 6.26258704707356015877913199355, 7.04739482916896435755820863629, 7.69283919060283494613682724389, 8.122888926522454399718102217182, 8.549240722330138415006197020084, 9.100419443419261215797783365286, 9.659062125287674825647078768492, 10.27471088867188668939987276364

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