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Properties

Label 4-1134e2-1.1-c1e2-0-86
Degree $4$
Conductor $1285956$
Sign $1$
Analytic cond. $81.9936$
Root an. cond. $3.00915$
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-s − 4·7-s + 8-s + 4·13-s + 4·14-s − 16-s − 6·17-s − 2·19-s − 6·23-s − 10·25-s − 4·26-s + 6·29-s − 5·31-s + 6·34-s − 8·37-s + 2·38-s − 3·41-s − 2·43-s + 6·46-s + 3·47-s + 9·49-s + 10·50-s − 6·53-s − 4·56-s − 6·58-s − 12·59-s − 8·61-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.51·7-s + 0.353·8-s + 1.10·13-s + 1.06·14-s − 1/4·16-s − 1.45·17-s − 0.458·19-s − 1.25·23-s − 2·25-s − 0.784·26-s + 1.11·29-s − 0.898·31-s + 1.02·34-s − 1.31·37-s + 0.324·38-s − 0.468·41-s − 0.304·43-s + 0.884·46-s + 0.437·47-s + 9/7·49-s + 1.41·50-s − 0.824·53-s − 0.534·56-s − 0.787·58-s − 1.56·59-s − 1.02·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1285956\)    =    \(2^{2} \cdot 3^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(81.9936\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1285956,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad2$C_2$ \( 1 + T + T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2^2$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_d
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_t
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_ap
23$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.23.g_cd
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_h
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.31.f_ag
37$C_2^2$ \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.37.i_bb
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.41.d_abg
43$C_2^2$ \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.43.c_abn
47$C_2^2$ \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.47.ad_abm
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_ar
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_dh
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_d
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_ad
71$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \) 2.71.be_od
73$C_2^2$ \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.73.l_bw
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) 2.79.ab_ada
83$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.83.a_adf
89$C_2^2$ \( 1 + 9 T - 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.89.j_ai
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.97.c_adp
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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

−9.415507279830890488392113325475, −9.321625853821069835027506088770, −8.704513245761455668575784358552, −8.671512861062864257852877442902, −7.984718837291666890931885437457, −7.69427999619948725340724044947, −6.97326153982318964600085723884, −6.85029001526945372571471571174, −6.08853665698931840005713745024, −6.04194366736133261960039293027, −5.66320204799691869378309545078, −4.64545205436501340054512287153, −4.30772461514516820171622143200, −3.89731672134636019352475857981, −3.25213847503822875344463596991, −2.87457999811761605231113501978, −1.88626956593863709125787110557, −1.60666715479128206862102868690, 0, 0, 1.60666715479128206862102868690, 1.88626956593863709125787110557, 2.87457999811761605231113501978, 3.25213847503822875344463596991, 3.89731672134636019352475857981, 4.30772461514516820171622143200, 4.64545205436501340054512287153, 5.66320204799691869378309545078, 6.04194366736133261960039293027, 6.08853665698931840005713745024, 6.85029001526945372571471571174, 6.97326153982318964600085723884, 7.69427999619948725340724044947, 7.984718837291666890931885437457, 8.671512861062864257852877442902, 8.704513245761455668575784358552, 9.321625853821069835027506088770, 9.415507279830890488392113325475

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