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Properties

Label 4-9522e2-1.1-c1e2-0-13
Degree $4$
Conductor $90668484$
Sign $1$
Analytic cond. $5781.10$
Root an. cond. $8.71972$
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 + 3·4-s − 2·5-s − 4·7-s + 4·8-s − 4·10-s + 2·11-s − 8·14-s + 5·16-s + 4·17-s − 6·19-s − 6·20-s + 4·22-s − 12·28-s − 8·29-s + 8·31-s + 6·32-s + 8·34-s + 8·35-s + 2·37-s − 12·38-s − 8·40-s − 12·41-s − 6·43-s + 6·44-s − 12·47-s − 2·49-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.894·5-s − 1.51·7-s + 1.41·8-s − 1.26·10-s + 0.603·11-s − 2.13·14-s + 5/4·16-s + 0.970·17-s − 1.37·19-s − 1.34·20-s + 0.852·22-s − 2.26·28-s − 1.48·29-s + 1.43·31-s + 1.06·32-s + 1.37·34-s + 1.35·35-s + 0.328·37-s − 1.94·38-s − 1.26·40-s − 1.87·41-s − 0.914·43-s + 0.904·44-s − 1.75·47-s − 2/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90668484 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90668484 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(90668484\)    =    \(2^{2} \cdot 3^{4} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(5781.10\)
Root analytic conductor: \(8.71972\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 90668484,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
23 \( 1 \)
good5$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_e
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
11$D_{4}$ \( 1 - 2 T + 16 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_q
13$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.13.a_ac
17$D_{4}$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_k
19$D_{4}$ \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.19.g_bo
29$D_{4}$ \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.29.i_bu
31$D_{4}$ \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.31.ai_by
37$D_{4}$ \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.37.ac_m
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$D_{4}$ \( 1 + 6 T + 88 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.43.g_dk
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.47.m_fa
53$D_{4}$ \( 1 - 2 T + 100 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.53.ac_dw
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$D_{4}$ \( 1 + 6 T + 124 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.61.g_eu
67$D_{4}$ \( 1 - 14 T + 120 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.67.ao_eq
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.71.am_gw
73$D_{4}$ \( 1 + 4 T + 122 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.73.e_es
79$D_{4}$ \( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.79.ae_by
83$D_{4}$ \( 1 + 22 T + 280 T^{2} + 22 p T^{3} + p^{2} T^{4} \) 2.83.w_ku
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$D_{4}$ \( 1 + 8 T + 182 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.97.i_ha
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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

−7.24515776565207922372967763351, −7.00250064876291098976375330068, −6.65488293176757595381630936310, −6.62736674283844275989223017120, −6.15259847368495097183841229619, −5.79960713410923103516763302826, −5.54830292795432285644473930720, −5.00910843941488530672493320934, −4.57037203427772143971788940150, −4.51844857312380677350214928843, −3.79788563375317648573894904528, −3.67543712045665851254515389605, −3.26808122581276878806932088356, −3.25689697812431168168648625385, −2.52986271483881353465043552105, −2.17444067041215523977959673401, −1.56327446107957143644344906927, −1.13354214469691637143842240064, 0, 0, 1.13354214469691637143842240064, 1.56327446107957143644344906927, 2.17444067041215523977959673401, 2.52986271483881353465043552105, 3.25689697812431168168648625385, 3.26808122581276878806932088356, 3.67543712045665851254515389605, 3.79788563375317648573894904528, 4.51844857312380677350214928843, 4.57037203427772143971788940150, 5.00910843941488530672493320934, 5.54830292795432285644473930720, 5.79960713410923103516763302826, 6.15259847368495097183841229619, 6.62736674283844275989223017120, 6.65488293176757595381630936310, 7.00250064876291098976375330068, 7.24515776565207922372967763351

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