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Properties

Label 4-3200e2-1.1-c1e2-0-34
Degree $4$
Conductor $10240000$
Sign $1$
Analytic cond. $652.911$
Root an. cond. $5.05491$
Motivic weight $1$
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 + 4·7-s − 9-s − 2·11-s + 6·17-s − 6·19-s + 8·21-s + 4·23-s − 6·27-s + 16·29-s + 4·31-s − 4·33-s + 4·37-s + 10·41-s + 20·43-s + 8·47-s + 6·49-s + 12·51-s − 4·53-s − 12·57-s − 4·59-s + 12·61-s − 4·63-s − 14·67-s + 8·69-s − 8·71-s − 6·73-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.51·7-s − 1/3·9-s − 0.603·11-s + 1.45·17-s − 1.37·19-s + 1.74·21-s + 0.834·23-s − 1.15·27-s + 2.97·29-s + 0.718·31-s − 0.696·33-s + 0.657·37-s + 1.56·41-s + 3.04·43-s + 1.16·47-s + 6/7·49-s + 1.68·51-s − 0.549·53-s − 1.58·57-s − 0.520·59-s + 1.53·61-s − 0.503·63-s − 1.71·67-s + 0.963·69-s − 0.949·71-s − 0.702·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10240000\)    =    \(2^{14} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(652.911\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10240000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.111726090\)
\(L(\frac12)\) \(\approx\) \(6.111726090\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_f
7$C_4$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.7.ae_k
11$D_{4}$ \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_f
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.13.a_ag
17$D_{4}$ \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_bj
19$D_{4}$ \( 1 + 6 T + 45 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.19.g_bt
23$C_4$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_bq
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.29.aq_es
31$D_{4}$ \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.31.ae_cg
37$D_{4}$ \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.37.ae_bu
41$D_{4}$ \( 1 - 10 T + 75 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.41.ak_cx
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.43.au_he
47$D_{4}$ \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.47.ai_da
53$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.53.e_da
59$D_{4}$ \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_dm
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.61.am_gc
67$D_{4}$ \( 1 + 14 T + 165 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.67.o_gj
71$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.71.i_ew
73$D_{4}$ \( 1 + 6 T + 147 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.73.g_fr
79$D_{4}$ \( 1 - 20 T + 250 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.79.au_jq
83$D_{4}$ \( 1 + 6 T + 157 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.83.g_gb
89$D_{4}$ \( 1 - 18 T + 251 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.89.as_jr
97$D_{4}$ \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.97.am_dy
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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

−8.733442805360386047773070057264, −8.512390981721874779757555629023, −8.002885861680480060099645078534, −7.908517769372263156576389110181, −7.57020398987145349790689460398, −7.30817477568889959544913289779, −6.41728991618517542853938408251, −6.34551529876477031325507692169, −5.72040429426910827415109590749, −5.50835844302553883161865425778, −4.82853920027608146479707005684, −4.61797870025230947504774040793, −4.23928366970178028955581235898, −3.77900661239747590984609846829, −2.91587557572118846892083833745, −2.86911077703104185558990978449, −2.45643542392718370575384806258, −2.01049395940819560208880248848, −1.00551589642208909555918666507, −0.899314277868807339486377108190, 0.899314277868807339486377108190, 1.00551589642208909555918666507, 2.01049395940819560208880248848, 2.45643542392718370575384806258, 2.86911077703104185558990978449, 2.91587557572118846892083833745, 3.77900661239747590984609846829, 4.23928366970178028955581235898, 4.61797870025230947504774040793, 4.82853920027608146479707005684, 5.50835844302553883161865425778, 5.72040429426910827415109590749, 6.34551529876477031325507692169, 6.41728991618517542853938408251, 7.30817477568889959544913289779, 7.57020398987145349790689460398, 7.908517769372263156576389110181, 8.002885861680480060099645078534, 8.512390981721874779757555629023, 8.733442805360386047773070057264

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