Properties

Label 4-338688-1.1-c1e2-0-51
Degree $4$
Conductor $338688$
Sign $-1$
Analytic cond. $21.5950$
Root an. cond. $2.15570$
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  − 3-s + 9-s − 4·13-s − 2·25-s − 27-s + 8·31-s − 12·37-s + 4·39-s + 49-s + 4·61-s + 8·67-s − 4·73-s + 2·75-s + 8·79-s + 81-s − 8·93-s − 20·97-s + 8·103-s − 4·109-s + 12·111-s − 4·117-s − 2·121-s + 127-s + 131-s + 137-s + 139-s − 147-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.10·13-s − 2/5·25-s − 0.192·27-s + 1.43·31-s − 1.97·37-s + 0.640·39-s + 1/7·49-s + 0.512·61-s + 0.977·67-s − 0.468·73-s + 0.230·75-s + 0.900·79-s + 1/9·81-s − 0.829·93-s − 2.03·97-s + 0.788·103-s − 0.383·109-s + 1.13·111-s − 0.369·117-s − 0.181·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0824·147-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(338688\)    =    \(2^{8} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(21.5950\)
Root analytic conductor: \(2.15570\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 338688,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
83$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−8.675418423468709114358813166365, −7.985115242388697754644488193749, −7.53675772484481950589551188116, −7.12228585343360915038815988604, −6.60631488437240327896776341806, −6.25439603531898400960762258129, −5.58338497298008639034308696011, −5.06166666791751219289703521050, −4.83560077083145676884113389525, −4.06277673546073422945848631930, −3.57842104901025308206047079084, −2.73483248161001198071408258224, −2.17595356787753299003691494949, −1.22827032572657991835025447717, 0, 1.22827032572657991835025447717, 2.17595356787753299003691494949, 2.73483248161001198071408258224, 3.57842104901025308206047079084, 4.06277673546073422945848631930, 4.83560077083145676884113389525, 5.06166666791751219289703521050, 5.58338497298008639034308696011, 6.25439603531898400960762258129, 6.60631488437240327896776341806, 7.12228585343360915038815988604, 7.53675772484481950589551188116, 7.985115242388697754644488193749, 8.675418423468709114358813166365

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