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Properties

Label 4-448e2-1.1-c1e2-0-26
Degree $4$
Conductor $200704$
Sign $-1$
Analytic cond. $12.7970$
Root an. cond. $1.89137$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s − 4·9-s + 2·11-s + 8·23-s − 4·25-s + 2·29-s − 14·37-s + 6·43-s − 3·49-s − 2·53-s + 8·63-s − 14·67-s − 12·71-s − 4·77-s + 16·79-s + 7·81-s − 8·99-s − 10·107-s + 6·109-s − 8·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 0.755·7-s − 4/3·9-s + 0.603·11-s + 1.66·23-s − 4/5·25-s + 0.371·29-s − 2.30·37-s + 0.914·43-s − 3/7·49-s − 0.274·53-s + 1.00·63-s − 1.71·67-s − 1.42·71-s − 0.455·77-s + 1.80·79-s + 7/9·81-s − 0.804·99-s − 0.966·107-s + 0.574·109-s − 0.752·113-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(200704\)    =    \(2^{12} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(12.7970\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 200704,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
7$C_2$ \( 1 + 2 T + p T^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.5.a_e
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.ac_o
13$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.13.a_e
17$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.17.a_ak
19$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.19.a_u
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.23.ai_bu
29$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.ac_k
31$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.31.a_abq
37$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.o_es
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.41.a_bi
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.43.ag_dq
47$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.47.a_ak
53$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.c_ec
59$C_2^2$ \( 1 - 76 T^{2} + p^{2} T^{4} \) 2.59.a_acy
61$C_2^2$ \( 1 + 52 T^{2} + p^{2} T^{4} \) 2.61.a_ca
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.67.o_gs
71$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.71.m_ek
73$C_2^2$ \( 1 + 106 T^{2} + p^{2} T^{4} \) 2.73.a_ec
79$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.79.aq_ik
83$C_2^2$ \( 1 - 76 T^{2} + p^{2} T^{4} \) 2.83.a_acy
89$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.89.a_acc
97$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.97.a_w
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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

−8.887359545196304099484904925506, −8.638361035804768482259583366102, −7.83909227598610422557414657660, −7.46177519197212909096581217426, −6.80556290473534560865696341437, −6.44009600108618296906958642497, −5.97729441297205905806774808887, −5.36107620290136741018353422503, −4.98827781379714457214623966723, −4.18039420334984038326262975775, −3.47034852288868160363314980887, −3.09694402156436104391404520916, −2.43458424726861079488953110546, −1.36823413058364660960900109756, 0, 1.36823413058364660960900109756, 2.43458424726861079488953110546, 3.09694402156436104391404520916, 3.47034852288868160363314980887, 4.18039420334984038326262975775, 4.98827781379714457214623966723, 5.36107620290136741018353422503, 5.97729441297205905806774808887, 6.44009600108618296906958642497, 6.80556290473534560865696341437, 7.46177519197212909096581217426, 7.83909227598610422557414657660, 8.638361035804768482259583366102, 8.887359545196304099484904925506

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