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Properties

Label 4-351e2-1.1-c1e2-0-20
Degree $4$
Conductor $123201$
Sign $-1$
Analytic cond. $7.85540$
Root an. cond. $1.67414$
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  − 4-s + 2·13-s − 3·16-s − 3·23-s − 25-s − 15·29-s − 8·43-s + 2·49-s − 2·52-s − 5·61-s + 7·64-s + 7·79-s + 3·92-s + 100-s − 6·101-s + 7·103-s − 18·107-s − 12·113-s + 15·116-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.554·13-s − 3/4·16-s − 0.625·23-s − 1/5·25-s − 2.78·29-s − 1.21·43-s + 2/7·49-s − 0.277·52-s − 0.640·61-s + 7/8·64-s + 0.787·79-s + 0.312·92-s + 1/10·100-s − 0.597·101-s + 0.689·103-s − 1.74·107-s − 1.12·113-s + 1.39·116-s + 8/11·121-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 123201 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123201 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(123201\)    =    \(3^{6} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(7.85540\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 123201,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad3 \( 1 \)
13$C_2$ \( 1 - 2 T + p T^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.2.a_b
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.11.a_ai
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.a_z
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.d_bc
29$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.29.p_ei
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.31.a_cg
37$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.37.a_ar
41$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \) 2.41.a_acn
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.47.a_bu
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.61.f_ee
67$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.67.a_bu
71$C_2^2$ \( 1 + 112 T^{2} + p^{2} T^{4} \) 2.71.a_ei
73$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \) 2.73.a_al
79$C_2$$\times$$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.ah_am
83$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \) 2.83.a_cm
89$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.89.a_w
97$C_2^2$ \( 1 + 133 T^{2} + p^{2} T^{4} \) 2.97.a_fd
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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

−9.105675264521832381983653615570, −8.899702461015159206725562790396, −8.122768408265860444734454425224, −7.83892606478377834130552153706, −7.23361567491057064981729992851, −6.66218270619862247635351487382, −6.17581367096526111799604391772, −5.47922211471740807769800147288, −5.18740085196123923769509184536, −4.31482067778899198564235868564, −3.90709927363477401840366602672, −3.33073065016862403926778738980, −2.32140469697422943102215446629, −1.58515090935433547270806176250, 0, 1.58515090935433547270806176250, 2.32140469697422943102215446629, 3.33073065016862403926778738980, 3.90709927363477401840366602672, 4.31482067778899198564235868564, 5.18740085196123923769509184536, 5.47922211471740807769800147288, 6.17581367096526111799604391772, 6.66218270619862247635351487382, 7.23361567491057064981729992851, 7.83892606478377834130552153706, 8.122768408265860444734454425224, 8.899702461015159206725562790396, 9.105675264521832381983653615570

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