Properties

Label 4-42e4-1.1-c3e2-0-2
Degree $4$
Conductor $3111696$
Sign $1$
Analytic cond. $10832.5$
Root an. cond. $10.2019$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 20·11-s − 8·13-s + 24·17-s − 44·19-s + 72·23-s + 125·25-s + 76·29-s − 184·31-s + 30·37-s + 432·41-s − 328·43-s + 520·47-s − 146·53-s − 80·55-s + 460·59-s − 628·61-s − 32·65-s − 556·67-s − 1.18e3·71-s − 1.02e3·73-s + 104·79-s + 648·83-s + 96·85-s + 896·89-s − 176·95-s − 1.84e3·97-s + ⋯
L(s)  = 1  + 0.357·5-s − 0.548·11-s − 0.170·13-s + 0.342·17-s − 0.531·19-s + 0.652·23-s + 25-s + 0.486·29-s − 1.06·31-s + 0.133·37-s + 1.64·41-s − 1.16·43-s + 1.61·47-s − 0.378·53-s − 0.196·55-s + 1.01·59-s − 1.31·61-s − 0.0610·65-s − 1.01·67-s − 1.97·71-s − 1.64·73-s + 0.148·79-s + 0.856·83-s + 0.122·85-s + 1.06·89-s − 0.190·95-s − 1.92·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3111696\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(10832.5\)
Root analytic conductor: \(10.2019\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3111696,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.402290514\)
\(L(\frac12)\) \(\approx\) \(1.402290514\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 - 4 T - 109 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 20 T - 931 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 4 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 24 T - 4337 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 44 T - 4923 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - 72 T - 6983 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 38 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 184 T + 4065 T^{2} + 184 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 30 T - 49753 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 216 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 164 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 520 T + 166577 T^{2} - 520 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 146 T - 127561 T^{2} + 146 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 460 T + 6221 T^{2} - 460 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 628 T + 167403 T^{2} + 628 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 556 T + 8373 T^{2} + 556 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 592 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 1024 T + 659559 T^{2} + 1024 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 104 T - 482223 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 - 324 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 896 T + 97847 T^{2} - 896 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 920 T + p^{3} T^{2} )^{2} \)
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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

−9.258932197644251308061130335448, −8.742950490049630464892116059117, −8.458925523683063962221149853482, −7.899180096897186414646517333749, −7.52868337721940108961666650486, −7.13776326290187642928562331267, −6.90727953252955168851646224053, −6.11662887089841813714241445136, −6.04874153282798302642696283570, −5.48929651679391181385867323296, −5.10640391310267851065185162363, −4.54187640471385713038089645722, −4.34897948332733520431332275877, −3.59921184502487452881769679229, −3.16420049115166233289312811920, −2.56794576055729228163834853399, −2.36393009003872537028210473111, −1.44211578610326664020393202519, −1.14065376453333108388021751581, −0.26253422742641477185379264064, 0.26253422742641477185379264064, 1.14065376453333108388021751581, 1.44211578610326664020393202519, 2.36393009003872537028210473111, 2.56794576055729228163834853399, 3.16420049115166233289312811920, 3.59921184502487452881769679229, 4.34897948332733520431332275877, 4.54187640471385713038089645722, 5.10640391310267851065185162363, 5.48929651679391181385867323296, 6.04874153282798302642696283570, 6.11662887089841813714241445136, 6.90727953252955168851646224053, 7.13776326290187642928562331267, 7.52868337721940108961666650486, 7.899180096897186414646517333749, 8.458925523683063962221149853482, 8.742950490049630464892116059117, 9.258932197644251308061130335448

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