Properties

Label 4-1856e2-1.1-c3e2-0-2
Degree $4$
Conductor $3444736$
Sign $1$
Analytic cond. $11991.9$
Root an. cond. $10.4645$
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  + 2·3-s + 10·5-s − 16·7-s − 45·9-s + 90·11-s + 50·13-s + 20·15-s − 44·17-s − 108·19-s − 32·21-s − 28·23-s + 41·25-s − 134·27-s − 58·29-s + 66·31-s + 180·33-s − 160·35-s − 40·37-s + 100·39-s + 304·41-s + 130·43-s − 450·45-s − 514·47-s − 110·49-s − 88·51-s + 958·53-s + 900·55-s + ⋯
L(s)  = 1  + 0.384·3-s + 0.894·5-s − 0.863·7-s − 5/3·9-s + 2.46·11-s + 1.06·13-s + 0.344·15-s − 0.627·17-s − 1.30·19-s − 0.332·21-s − 0.253·23-s + 0.327·25-s − 0.955·27-s − 0.371·29-s + 0.382·31-s + 0.949·33-s − 0.772·35-s − 0.177·37-s + 0.410·39-s + 1.15·41-s + 0.461·43-s − 1.49·45-s − 1.59·47-s − 0.320·49-s − 0.241·51-s + 2.48·53-s + 2.20·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3444736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3444736 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3444736\)    =    \(2^{12} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(11991.9\)
Root analytic conductor: \(10.4645\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1856} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3444736,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.932426787\)
\(L(\frac12)\) \(\approx\) \(3.932426787\)
\(L(\frac{5}{2})\) 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 \( 1 \)
29$C_1$ \( ( 1 + p T )^{2} \)
good3$D_{4}$ \( 1 - 2 T + 49 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \)
5$D_{4}$ \( 1 - 2 p T + 59 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 + 16 T + 366 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 90 T + 4633 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 50 T + 4635 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 44 T + 8774 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 108 T + 16538 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 28 T - 11974 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 66 T - 10615 T^{2} - 66 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 40 T + 101610 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 304 T + 122546 T^{2} - 304 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 130 T + 146385 T^{2} - 130 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 514 T + 214889 T^{2} + 514 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 958 T + 510971 T^{2} - 958 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 180 T + 43858 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 1028 T + 717294 T^{2} + 1028 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 912 T + 807062 T^{2} - 912 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 796 T + 867290 T^{2} - 796 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 856 T + 775362 T^{2} + 856 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 318 T + 229433 T^{2} + 318 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 1828 T + 1970306 T^{2} - 1828 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 944 T + 1601618 T^{2} + 944 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 368 T + 1799202 T^{2} - 368 p^{3} T^{3} + p^{6} T^{4} \)
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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

−9.073610711614418324499108083097, −8.769479737751325647048835913566, −8.412495514299114916017531901731, −8.244756372905439831809539320694, −7.39741431500271757074391105961, −7.00556073560597870071801035131, −6.35809910755019332423044222800, −6.31928024839069932214993420526, −6.10804389688195153774177041288, −5.72742565749569061642830735776, −5.06715942564098621447472922715, −4.45592025332053249081284439596, −3.96169443977272417702898138007, −3.65226383290654908420890471325, −3.22588952422120851375556921165, −2.67342354536231267214817265604, −2.01156523622587117318376398948, −1.82960640149067584448321496804, −0.924704832233745873209505789863, −0.46397687471829912936621347208, 0.46397687471829912936621347208, 0.924704832233745873209505789863, 1.82960640149067584448321496804, 2.01156523622587117318376398948, 2.67342354536231267214817265604, 3.22588952422120851375556921165, 3.65226383290654908420890471325, 3.96169443977272417702898138007, 4.45592025332053249081284439596, 5.06715942564098621447472922715, 5.72742565749569061642830735776, 6.10804389688195153774177041288, 6.31928024839069932214993420526, 6.35809910755019332423044222800, 7.00556073560597870071801035131, 7.39741431500271757074391105961, 8.244756372905439831809539320694, 8.412495514299114916017531901731, 8.769479737751325647048835913566, 9.073610711614418324499108083097

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