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Properties

Label 4-448e2-1.1-c1e2-0-31
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 + 2·9-s − 2·11-s − 12·23-s + 2·25-s + 12·29-s − 16·37-s + 6·43-s − 3·49-s − 4·53-s − 4·63-s − 6·67-s − 12·71-s + 4·77-s − 5·81-s − 4·99-s + 6·107-s − 4·109-s − 12·113-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 0.755·7-s + 2/3·9-s − 0.603·11-s − 2.50·23-s + 2/5·25-s + 2.22·29-s − 2.63·37-s + 0.914·43-s − 3/7·49-s − 0.549·53-s − 0.503·63-s − 0.733·67-s − 1.42·71-s + 0.455·77-s − 5/9·81-s − 0.402·99-s + 0.580·107-s − 0.383·109-s − 1.12·113-s − 1.63·121-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 + 0.0798·157-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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 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$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.c_w
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.17.a_ao
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.m_da
29$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.29.am_da
31$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.31.a_abe
37$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.q_fe
41$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.41.a_ac
43$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.ag_bu
47$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.47.a_abu
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.e_dq
59$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \) 2.59.a_adu
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.61.a_acg
67$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.g_fm
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.71.m_gw
73$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.73.a_cw
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.a_cg
83$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.83.a_abi
89$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.89.a_bq
97$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.97.a_bi
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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

−8.746524964287837260452019421604, −8.401247136974066491031045692871, −7.964160535307739886485964146967, −7.34866174415253259695739288264, −6.95776410404516941809188010248, −6.38002916034409695923504060740, −6.02980111525064099969572226310, −5.41843571596296885019384326168, −4.72834457742773847842582342101, −4.30804269874971598210063051924, −3.61629865853342758934189995020, −3.03423176576253579061835864271, −2.30748650040207537967845148195, −1.45805955405842403317092570268, 0, 1.45805955405842403317092570268, 2.30748650040207537967845148195, 3.03423176576253579061835864271, 3.61629865853342758934189995020, 4.30804269874971598210063051924, 4.72834457742773847842582342101, 5.41843571596296885019384326168, 6.02980111525064099969572226310, 6.38002916034409695923504060740, 6.95776410404516941809188010248, 7.34866174415253259695739288264, 7.964160535307739886485964146967, 8.401247136974066491031045692871, 8.746524964287837260452019421604

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