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Properties

Label 4-1134e2-1.1-c1e2-0-13
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-s − 6·5-s − 4·7-s + 8-s + 6·10-s − 6·11-s + 4·13-s + 4·14-s − 16-s + 4·19-s + 6·22-s + 17·25-s − 4·26-s + 9·29-s + 31-s + 24·35-s − 8·37-s − 4·38-s − 6·40-s + 10·43-s − 6·47-s + 9·49-s − 17·50-s − 3·53-s + 36·55-s − 4·56-s − 9·58-s + ⋯
L(s)  = 1  − 0.707·2-s − 2.68·5-s − 1.51·7-s + 0.353·8-s + 1.89·10-s − 1.80·11-s + 1.10·13-s + 1.06·14-s − 1/4·16-s + 0.917·19-s + 1.27·22-s + 17/5·25-s − 0.784·26-s + 1.67·29-s + 0.179·31-s + 4.05·35-s − 1.31·37-s − 0.648·38-s − 0.948·40-s + 1.52·43-s − 0.875·47-s + 9/7·49-s − 2.40·50-s − 0.412·53-s + 4.85·55-s − 0.534·56-s − 1.18·58-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

\(L(1)\) \(\approx\) \(0.3484194328\)
\(L(\frac12)\) \(\approx\) \(0.3484194328\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.5.g_t
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.11.g_bf
13$C_2^2$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_d
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_ad
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2^2$ \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.29.aj_ca
31$C_2^2$ \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) 2.31.ab_abe
37$C_2^2$ \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.37.i_bb
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.41.a_abp
43$C_2^2$ \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.43.ak_cf
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_al
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.53.d_abs
59$C_2^2$ \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.59.ad_aby
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.61.ak_bn
67$C_2^2$ \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.67.ak_bh
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.71.am_gw
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_acr
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) 2.79.ab_ada
83$C_2^2$ \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.83.j_ac
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.89.ag_acb
97$C_2^2$ \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \) 2.97.ab_ads
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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

−10.11037854145630395391381806303, −9.653612908057771892577354172587, −9.015294700930763744891030780614, −8.744369563981517876634704711525, −8.205699823767514146455640557572, −8.075290885194319396111210317419, −7.68951186471539957170931728713, −7.28234333246632456654464533604, −6.90395084743673578452748678067, −6.43419274011781789983994184229, −5.89901429984915536176116827754, −5.14221082664712962715647889333, −4.91619895746372605971117492610, −4.13137034474540840853394773361, −3.81420342988359060376885150845, −3.28750892510202225909780605845, −3.07575075472159134007379977860, −2.31352085574566267005931063868, −0.867481232918765214860434098757, −0.41478198035576836475555357687, 0.41478198035576836475555357687, 0.867481232918765214860434098757, 2.31352085574566267005931063868, 3.07575075472159134007379977860, 3.28750892510202225909780605845, 3.81420342988359060376885150845, 4.13137034474540840853394773361, 4.91619895746372605971117492610, 5.14221082664712962715647889333, 5.89901429984915536176116827754, 6.43419274011781789983994184229, 6.90395084743673578452748678067, 7.28234333246632456654464533604, 7.68951186471539957170931728713, 8.075290885194319396111210317419, 8.205699823767514146455640557572, 8.744369563981517876634704711525, 9.015294700930763744891030780614, 9.653612908057771892577354172587, 10.11037854145630395391381806303

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