Properties

Label 4-1856e2-1.1-c3e2-0-0
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  + 10·5-s + 20·7-s − 41·9-s − 32·11-s + 54·13-s − 44·17-s − 32·19-s + 36·23-s − 123·25-s + 58·29-s − 20·31-s + 200·35-s + 144·37-s + 96·41-s + 240·43-s − 410·45-s − 596·47-s − 178·49-s + 34·53-s − 320·55-s + 724·59-s + 612·61-s − 820·63-s + 540·65-s + 528·67-s + 104·71-s − 872·73-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.07·7-s − 1.51·9-s − 0.877·11-s + 1.15·13-s − 0.627·17-s − 0.386·19-s + 0.326·23-s − 0.983·25-s + 0.371·29-s − 0.115·31-s + 0.965·35-s + 0.639·37-s + 0.365·41-s + 0.851·43-s − 1.35·45-s − 1.84·47-s − 0.518·49-s + 0.0881·53-s − 0.784·55-s + 1.59·59-s + 1.28·61-s − 1.63·63-s + 1.03·65-s + 0.962·67-s + 0.173·71-s − 1.39·73-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

\(L(2)\) \(\approx\) \(2.689825235\)
\(L(\frac12)\) \(\approx\) \(2.689825235\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 + 41 T^{2} + p^{6} T^{4} \)
5$D_{4}$ \( 1 - 2 p T + 223 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 - 20 T + 578 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 32 T + 2905 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 54 T + 4655 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 44 T + 4018 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 32 T + 12102 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 36 T + 23358 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 20 T + 59565 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 144 T + 76538 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 96 T + 140094 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 240 T + 172361 T^{2} - 240 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 596 T + 265237 T^{2} + 596 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 34 T + 260135 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 724 T + 478102 T^{2} - 724 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 612 T + 412346 T^{2} - 612 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 528 T + 130214 T^{2} - 528 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 104 T + 360298 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 872 T + 601218 T^{2} + 872 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 820 T + 800253 T^{2} - 820 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 228 T + 1051270 T^{2} + 228 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 32 T - 486818 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 1896 T + 2701118 T^{2} + 1896 p^{3} T^{3} + p^{6} T^{4} \)
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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

−8.803157496399321284793876712237, −8.717241747905845761285561613447, −8.287680416233651716697518654001, −8.116517138795061198679930222704, −7.61435556390224503058741327100, −7.17154985178421822366248165565, −6.42506496057378206032892381811, −6.28811726016516622683607964524, −5.88540284967537646962097905030, −5.44891607585531995770960775026, −5.08795598871894460223517123659, −4.81208146739665162395787704916, −4.00845322439387898969074545875, −3.78638405694683337089603308626, −2.94119380656423376600886343846, −2.68343679653185104039301018473, −2.02220794770628273548896061955, −1.81073669264476657940451618591, −0.980944108032293421638941687731, −0.37007931889644157044188192798, 0.37007931889644157044188192798, 0.980944108032293421638941687731, 1.81073669264476657940451618591, 2.02220794770628273548896061955, 2.68343679653185104039301018473, 2.94119380656423376600886343846, 3.78638405694683337089603308626, 4.00845322439387898969074545875, 4.81208146739665162395787704916, 5.08795598871894460223517123659, 5.44891607585531995770960775026, 5.88540284967537646962097905030, 6.28811726016516622683607964524, 6.42506496057378206032892381811, 7.17154985178421822366248165565, 7.61435556390224503058741327100, 8.116517138795061198679930222704, 8.287680416233651716697518654001, 8.717241747905845761285561613447, 8.803157496399321284793876712237

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