Properties

Label 4-464e2-1.1-c3e2-0-4
Degree $4$
Conductor $215296$
Sign $1$
Analytic cond. $749.493$
Root an. cond. $5.23229$
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  − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 215296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215296 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(215296\)    =    \(2^{8} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(749.493\)
Root analytic conductor: \(5.23229\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 215296,\ (\ :3/2, 3/2),\ 1)\)

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 \( 1 \)
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

−10.58122436366480021157989292174, −9.890985291080870011127910076927, −9.333012596095464937501338639715, −9.139605125996630549710038388398, −8.442331954645941093147449333367, −8.189590516800444070683602097383, −7.59451625528674353705592863978, −7.50997477847901691637331863397, −6.51176370568029203083249794954, −6.40514427635021372329151815231, −5.41021900236051326794866398169, −5.25469519476066271520501814355, −4.49594051411567214198295637670, −4.19549282936872331053748703961, −3.27524854804100708273559314699, −2.98055685404707266727496648212, −1.99346490825132851247852964442, −1.45731604287193425478972076083, 0, 0, 1.45731604287193425478972076083, 1.99346490825132851247852964442, 2.98055685404707266727496648212, 3.27524854804100708273559314699, 4.19549282936872331053748703961, 4.49594051411567214198295637670, 5.25469519476066271520501814355, 5.41021900236051326794866398169, 6.40514427635021372329151815231, 6.51176370568029203083249794954, 7.50997477847901691637331863397, 7.59451625528674353705592863978, 8.189590516800444070683602097383, 8.442331954645941093147449333367, 9.139605125996630549710038388398, 9.333012596095464937501338639715, 9.890985291080870011127910076927, 10.58122436366480021157989292174

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