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Properties

Label 4-448e2-1.1-c1e2-0-14
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·7-s + 2·9-s + 4·11-s − 8·23-s + 2·25-s − 4·29-s − 4·37-s + 4·43-s + 9·49-s + 12·53-s + 8·63-s + 12·67-s + 8·71-s + 16·77-s + 16·79-s − 5·81-s + 8·99-s + 4·107-s − 4·109-s − 12·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 1.51·7-s + 2/3·9-s + 1.20·11-s − 1.66·23-s + 2/5·25-s − 0.742·29-s − 0.657·37-s + 0.609·43-s + 9/7·49-s + 1.64·53-s + 1.00·63-s + 1.46·67-s + 0.949·71-s + 1.82·77-s + 1.80·79-s − 5/9·81-s + 0.804·99-s + 0.386·107-s − 0.383·109-s − 1.12·113-s − 0.545·121-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: \(0\)
Selberg data: \((4,\ 200704,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.329195233\)
\(L(\frac12)\) \(\approx\) \(2.329195233\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 4 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.11.ae_w
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.13.a_o
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.17.a_o
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.i_bu
29$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.e_bu
31$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.31.a_as
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.e_ck
41$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.41.a_o
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.ae_cc
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.53.am_ew
59$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.59.a_bu
61$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.61.a_ade
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.67.am_gk
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.71.ai_dq
73$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \) 2.73.a_dq
79$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.79.aq_hy
83$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \) 2.83.a_dq
89$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.89.a_aco
97$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.97.a_ade
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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

−9.011530917577971838679640259715, −8.613276852452012119087367726350, −8.121618991769280190033617141191, −7.69506373849898615837060537622, −7.26477051048943201028761904628, −6.65843415608608869717599728348, −6.27898075727928037077278809923, −5.46912232568141088593189111437, −5.19300823978676382926111286843, −4.42492522923214735887745097890, −3.98806710161747150010342947178, −3.64128701932627348216621737358, −2.37072722824809261902536430418, −1.84326165403814571098871892320, −1.08973477197510465215355246143, 1.08973477197510465215355246143, 1.84326165403814571098871892320, 2.37072722824809261902536430418, 3.64128701932627348216621737358, 3.98806710161747150010342947178, 4.42492522923214735887745097890, 5.19300823978676382926111286843, 5.46912232568141088593189111437, 6.27898075727928037077278809923, 6.65843415608608869717599728348, 7.26477051048943201028761904628, 7.69506373849898615837060537622, 8.121618991769280190033617141191, 8.613276852452012119087367726350, 9.011530917577971838679640259715

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