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Properties

Label 4-1001e2-1.1-c1e2-0-0
Degree $4$
Conductor $1002001$
Sign $-1$
Analytic cond. $63.8884$
Root an. cond. $2.82719$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·5-s − 6·9-s − 6·11-s − 4·16-s + 6·23-s + 17·25-s − 6·31-s − 8·37-s + 36·45-s + 14·47-s + 49-s − 18·53-s + 36·55-s + 16·59-s − 12·67-s − 16·71-s + 24·80-s + 27·81-s + 6·89-s + 14·97-s + 36·99-s − 8·103-s − 6·113-s − 36·115-s + 25·121-s − 18·125-s + 127-s + ⋯
L(s)  = 1  − 2.68·5-s − 2·9-s − 1.80·11-s − 16-s + 1.25·23-s + 17/5·25-s − 1.07·31-s − 1.31·37-s + 5.36·45-s + 2.04·47-s + 1/7·49-s − 2.47·53-s + 4.85·55-s + 2.08·59-s − 1.46·67-s − 1.89·71-s + 2.68·80-s + 3·81-s + 0.635·89-s + 1.42·97-s + 3.61·99-s − 0.788·103-s − 0.564·113-s − 3.35·115-s + 2.27·121-s − 1.60·125-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1002001 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1002001 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1002001\)    =    \(7^{2} \cdot 11^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(63.8884\)
Root analytic conductor: \(2.82719\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1002001,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_2$ \( 1 + 6 T + p T^{2} \)
13$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$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.5.g_t
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.19.a_n
23$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.23.ag_cd
29$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.29.a_bh
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.31.g_ct
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.37.i_dm
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.43.a_dh
47$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.47.ao_fn
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.53.s_hf
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.59.aq_ha
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.67.m_go
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.73.a_ax
79$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.79.a_ft
83$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.83.a_ach
89$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.89.ag_hf
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.97.ao_jj
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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

−7.88305646625584781075065151406, −7.43563929744309328324545511818, −7.38318915968245477906353475134, −6.75747750884296153616010532091, −6.10493668511615122876893578729, −5.41120349293735009167808988128, −5.27883313770925671699270459709, −4.64299598156704998773380783331, −4.22570906854938018403998871989, −3.57094110488123461480235960376, −3.11744276397134147631028514628, −2.85617269550579922001781563132, −2.11792864837681218638414366758, −0.49790947708643134938837315000, 0, 0.49790947708643134938837315000, 2.11792864837681218638414366758, 2.85617269550579922001781563132, 3.11744276397134147631028514628, 3.57094110488123461480235960376, 4.22570906854938018403998871989, 4.64299598156704998773380783331, 5.27883313770925671699270459709, 5.41120349293735009167808988128, 6.10493668511615122876893578729, 6.75747750884296153616010532091, 7.38318915968245477906353475134, 7.43563929744309328324545511818, 7.88305646625584781075065151406

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