Properties

Label 4-2352e2-1.1-c3e2-0-17
Degree $4$
Conductor $5531904$
Sign $1$
Analytic cond. $19257.8$
Root an. cond. $11.7801$
Motivic weight $3$
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  + 6·3-s − 6·5-s + 27·9-s + 6·11-s − 16·13-s − 36·15-s + 6·17-s + 64·19-s − 6·23-s − 166·25-s + 108·27-s − 252·29-s + 40·31-s + 36·33-s − 248·37-s − 96·39-s + 450·41-s − 376·43-s − 162·45-s − 12·47-s + 36·51-s − 1.10e3·53-s − 36·55-s + 384·57-s + 804·59-s + 428·61-s + 96·65-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.536·5-s + 9-s + 0.164·11-s − 0.341·13-s − 0.619·15-s + 0.0856·17-s + 0.772·19-s − 0.0543·23-s − 1.32·25-s + 0.769·27-s − 1.61·29-s + 0.231·31-s + 0.189·33-s − 1.10·37-s − 0.394·39-s + 1.71·41-s − 1.33·43-s − 0.536·45-s − 0.0372·47-s + 0.0988·51-s − 2.86·53-s − 0.0882·55-s + 0.892·57-s + 1.77·59-s + 0.898·61-s + 0.183·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5531904\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(19257.8\)
Root analytic conductor: \(11.7801\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5531904,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{2} \)
7 \( 1 \)
good5$D_{4}$ \( 1 + 6 T + 202 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 6 T + 1246 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 16 T + 2406 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 6 T + 9778 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 64 T + 6534 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 6 T + 7870 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 252 T + 56446 T^{2} + 252 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 40 T - 13890 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 248 T + 98214 T^{2} + 248 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 450 T + 175642 T^{2} - 450 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 376 T + 161526 T^{2} + 376 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 12 T + 141790 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 1104 T + 602230 T^{2} + 1104 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 804 T + 380614 T^{2} - 804 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 428 T + 425886 T^{2} - 428 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 148 T + 440790 T^{2} + 148 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 954 T + 13106 p T^{2} + 954 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 1072 T + 1063278 T^{2} + 1072 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 572 T + 901662 T^{2} - 572 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 1944 T + 1957030 T^{2} - 1944 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 366 T + 1156090 T^{2} + 366 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 808 T + 903054 T^{2} + 808 p^{3} T^{3} + p^{6} T^{4} \)
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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

−8.205758253550049286240659183678, −8.183708164598938593441898074661, −7.69333961758841370526986745340, −7.47905335322993365232125044789, −6.87022043742042983277787364342, −6.86178402921126138828018829689, −5.94821738587100335265558533688, −5.84294965280489941047744059433, −5.13849556633611255921516790337, −4.88422898062411965712946029257, −4.13703419761613135581947811739, −4.02007516798388619162543308264, −3.37897544399514458125701179119, −3.28202969547634036390091411165, −2.50627674639791056357748268712, −2.18595070889658692551586986470, −1.50019691474786066294585784667, −1.19484605574209766719975417736, 0, 0, 1.19484605574209766719975417736, 1.50019691474786066294585784667, 2.18595070889658692551586986470, 2.50627674639791056357748268712, 3.28202969547634036390091411165, 3.37897544399514458125701179119, 4.02007516798388619162543308264, 4.13703419761613135581947811739, 4.88422898062411965712946029257, 5.13849556633611255921516790337, 5.84294965280489941047744059433, 5.94821738587100335265558533688, 6.86178402921126138828018829689, 6.87022043742042983277787364342, 7.47905335322993365232125044789, 7.69333961758841370526986745340, 8.183708164598938593441898074661, 8.205758253550049286240659183678

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