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Properties

Label 4-1134e2-1.1-c1e2-0-66
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s + 5·7-s + 4·8-s + 4·13-s + 10·14-s + 5·16-s − 6·17-s − 2·19-s + 3·23-s + 5·25-s + 8·26-s + 15·28-s + 6·29-s + 10·31-s + 6·32-s − 12·34-s − 8·37-s − 4·38-s − 3·41-s − 2·43-s + 6·46-s − 6·47-s + 18·49-s + 10·50-s + 12·52-s − 6·53-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.88·7-s + 1.41·8-s + 1.10·13-s + 2.67·14-s + 5/4·16-s − 1.45·17-s − 0.458·19-s + 0.625·23-s + 25-s + 1.56·26-s + 2.83·28-s + 1.11·29-s + 1.79·31-s + 1.06·32-s − 2.05·34-s − 1.31·37-s − 0.648·38-s − 0.468·41-s − 0.304·43-s + 0.884·46-s − 0.875·47-s + 18/7·49-s + 1.41·50-s + 1.66·52-s − 0.824·53-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: \(0\)
Selberg data: \((4,\ 1285956,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.048210163\)
\(L(\frac12)\) \(\approx\) \(8.048210163\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.5.a_af
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.11.a_al
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$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.23.ad_ao
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_h
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.31.ak_dj
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$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.47.g_dz
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_ar
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.61.aq_he
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.67.aq_hq
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$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.79.c_gd
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

−10.03498125126303768559099391071, −10.02970648744646252098889400772, −8.747743310230295735259609702422, −8.689223489429598187327129036659, −8.433095102713595573631883100319, −8.129033338426144666038968600108, −7.20044604348840720991837768708, −7.13239069102917032161630753226, −6.44499856994458284274529736581, −6.39911864534848620749143303972, −5.57783172556145096946663709445, −5.18562359408352725886083039487, −4.75921206485804303885288800752, −4.56813176005885191995840177043, −3.96919643955467134984066752511, −3.57734765278609824741732005798, −2.55998803375001182095488532862, −2.54062012141973695952783034456, −1.55162904800395650213867468481, −1.14538182954629501756742180596, 1.14538182954629501756742180596, 1.55162904800395650213867468481, 2.54062012141973695952783034456, 2.55998803375001182095488532862, 3.57734765278609824741732005798, 3.96919643955467134984066752511, 4.56813176005885191995840177043, 4.75921206485804303885288800752, 5.18562359408352725886083039487, 5.57783172556145096946663709445, 6.39911864534848620749143303972, 6.44499856994458284274529736581, 7.13239069102917032161630753226, 7.20044604348840720991837768708, 8.129033338426144666038968600108, 8.433095102713595573631883100319, 8.689223489429598187327129036659, 8.747743310230295735259609702422, 10.02970648744646252098889400772, 10.03498125126303768559099391071

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