Properties

Label 4-2299e2-1.1-c1e2-0-5
Degree $4$
Conductor $5285401$
Sign $1$
Analytic cond. $337.001$
Root an. cond. $4.28457$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s + 6·5-s − 2·7-s − 4·8-s − 3·9-s − 12·10-s − 2·13-s + 4·14-s + 8·16-s − 6·17-s + 6·18-s − 2·19-s + 12·20-s + 6·23-s + 17·25-s + 4·26-s − 4·28-s − 14·29-s − 16·31-s − 8·32-s + 12·34-s − 12·35-s − 6·36-s + 8·37-s + 4·38-s − 24·40-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 2.68·5-s − 0.755·7-s − 1.41·8-s − 9-s − 3.79·10-s − 0.554·13-s + 1.06·14-s + 2·16-s − 1.45·17-s + 1.41·18-s − 0.458·19-s + 2.68·20-s + 1.25·23-s + 17/5·25-s + 0.784·26-s − 0.755·28-s − 2.59·29-s − 2.87·31-s − 1.41·32-s + 2.05·34-s − 2.02·35-s − 36-s + 1.31·37-s + 0.648·38-s − 3.79·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5285401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5285401 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5285401\)    =    \(11^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(337.001\)
Root analytic conductor: \(4.28457\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5285401,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad11 \( 1 \)
19$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) 2.2.c_c
3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.3.a_d
5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.5.ag_t
7$D_{4}$ \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_m
13$D_{4}$ \( 1 + 2 T + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_a
17$D_{4}$ \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_bo
23$D_{4}$ \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_br
29$D_{4}$ \( 1 + 14 T + 104 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.29.o_ea
31$D_{4}$ \( 1 + 16 T + 123 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.31.q_et
37$D_{4}$ \( 1 - 8 T + 87 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.37.ai_dj
41$D_{4}$ \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.41.i_by
43$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.43.ai_cc
47$D_{4}$ \( 1 + 16 T + 146 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.47.q_fq
53$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.53.a_cg
59$D_{4}$ \( 1 - 4 T + 47 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.59.ae_bv
61$D_{4}$ \( 1 + 2 T + 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.61.c_bw
67$D_{4}$ \( 1 - 4 T + 63 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_cl
71$D_{4}$ \( 1 + 20 T + 239 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.71.u_jf
73$D_{4}$ \( 1 + 20 T + 234 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.73.u_ja
79$D_{4}$ \( 1 - 14 T + 204 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.79.ao_hw
83$D_{4}$ \( 1 - 18 T + 220 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.83.as_im
89$D_{4}$ \( 1 - 4 T + 155 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.89.ae_fz
97$D_{4}$ \( 1 + 12 T + 203 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.97.m_hv
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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

−9.009202637911170171285771524900, −8.915178027263180578482754689770, −8.100775486981640863612640932353, −7.891947566320313383690594233000, −7.07878581638247806015693739992, −6.95127102571720772824303610389, −6.49348489161045076251591165072, −5.97467404756361919107582665659, −5.84581377127373516372025404645, −5.57927050368605344959000240086, −5.10793376135236964568338085770, −4.55337200035150109420351732718, −3.57861929139291241875667378311, −3.23633191001908139413348237573, −2.73304982507010235726556550648, −2.08500154715902063083720263430, −2.01134125403917145933850536854, −1.38190658442343646121241102472, 0, 0, 1.38190658442343646121241102472, 2.01134125403917145933850536854, 2.08500154715902063083720263430, 2.73304982507010235726556550648, 3.23633191001908139413348237573, 3.57861929139291241875667378311, 4.55337200035150109420351732718, 5.10793376135236964568338085770, 5.57927050368605344959000240086, 5.84581377127373516372025404645, 5.97467404756361919107582665659, 6.49348489161045076251591165072, 6.95127102571720772824303610389, 7.07878581638247806015693739992, 7.891947566320313383690594233000, 8.100775486981640863612640932353, 8.915178027263180578482754689770, 9.009202637911170171285771524900

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