Properties

Label 4-2299e2-1.1-c1e2-0-6
Degree $4$
Conductor $5285401$
Sign $-1$
Analytic cond. $337.001$
Root an. cond. $4.28457$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s − 6·5-s + 21·9-s + 36·15-s − 4·16-s + 6·23-s + 17·25-s − 54·27-s + 2·31-s − 22·37-s − 126·45-s − 16·47-s + 24·48-s − 10·49-s − 12·53-s − 2·59-s − 26·67-s − 36·69-s − 26·71-s − 102·75-s + 24·80-s + 108·81-s − 14·89-s − 12·93-s − 18·97-s − 16·103-s + 132·111-s + ⋯
L(s)  = 1  − 3.46·3-s − 2.68·5-s + 7·9-s + 9.29·15-s − 16-s + 1.25·23-s + 17/5·25-s − 10.3·27-s + 0.359·31-s − 3.61·37-s − 18.7·45-s − 2.33·47-s + 3.46·48-s − 1.42·49-s − 1.64·53-s − 0.260·59-s − 3.17·67-s − 4.33·69-s − 3.08·71-s − 11.7·75-s + 2.68·80-s + 12·81-s − 1.48·89-s − 1.24·93-s − 1.82·97-s − 1.57·103-s + 12.5·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5285401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5285401 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5285401\)    =    \(11^{4} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(337.001\)
Root analytic conductor: \(4.28457\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((4,\ 5285401,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad11 \( 1 \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.2.a_a
3$C_2$ \( ( 1 + p T + p T^{2} )^{2} \) 2.3.g_p
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.5.g_t
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
23$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.23.ag_cd
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.31.ac_cl
37$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.37.w_hn
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.a_s
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.47.q_gc
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.59.c_ep
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.a_acw
67$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.67.ba_lr
71$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.71.ba_lz
73$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.73.a_fa
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.a_cg
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.89.o_it
97$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.97.s_kp
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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

−6.83058479316239850878642874578, −6.32445094824772140645055180741, −5.84131011128128071057838574907, −5.42961123901025340308116063784, −4.84709984478174556407926642734, −4.67192305890313338178620814169, −4.54900748914880908361739071157, −4.02384769932422906633259041202, −3.29418866089581917859982179260, −3.15719648166913061071857116095, −1.45072784072277250963527793735, −1.43058089842377162335136228237, 0, 0, 0, 1.43058089842377162335136228237, 1.45072784072277250963527793735, 3.15719648166913061071857116095, 3.29418866089581917859982179260, 4.02384769932422906633259041202, 4.54900748914880908361739071157, 4.67192305890313338178620814169, 4.84709984478174556407926642734, 5.42961123901025340308116063784, 5.84131011128128071057838574907, 6.32445094824772140645055180741, 6.83058479316239850878642874578

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