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Properties

Label 4-1040e2-1.1-c1e2-0-49
Degree $4$
Conductor $1081600$
Sign $-1$
Analytic cond. $68.9637$
Root an. cond. $2.88174$
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  − 2·5-s − 2·9-s − 2·13-s + 3·25-s + 4·45-s − 2·49-s + 12·53-s − 8·61-s + 4·65-s − 5·81-s − 12·89-s + 24·97-s + 12·101-s + 24·109-s − 12·113-s + 4·117-s + 10·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 0.894·5-s − 2/3·9-s − 0.554·13-s + 3/5·25-s + 0.596·45-s − 2/7·49-s + 1.64·53-s − 1.02·61-s + 0.496·65-s − 5/9·81-s − 1.27·89-s + 2.43·97-s + 1.19·101-s + 2.29·109-s − 1.12·113-s + 0.369·117-s + 0.909·121-s − 0.357·125-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 + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1081600\)    =    \(2^{8} \cdot 5^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(68.9637\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1081600,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.19.a_o
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.23.a_ac
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.31.a_abu
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.a_aba
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.43.a_bi
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.59.a_ck
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.i_dy
67$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.67.a_aw
71$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.71.a_ba
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.79.a_aby
83$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.83.a_by
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) 2.97.ay_mw
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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

−7.80019523012963931903075954936, −7.31280066535498996112985725664, −7.26927111972768690986858989426, −6.55596474111111648673289672386, −6.05877445646593256691786933704, −5.70242119179958153730749881149, −5.05114780090473010136074821908, −4.71246525005282268147051099678, −4.19328938157817020898215093129, −3.60778916273215204704984402016, −3.17849108100061118271871533883, −2.58295993891423193456421003265, −1.97520006840294894021148545522, −0.942726694253668839126892786514, 0, 0.942726694253668839126892786514, 1.97520006840294894021148545522, 2.58295993891423193456421003265, 3.17849108100061118271871533883, 3.60778916273215204704984402016, 4.19328938157817020898215093129, 4.71246525005282268147051099678, 5.05114780090473010136074821908, 5.70242119179958153730749881149, 6.05877445646593256691786933704, 6.55596474111111648673289672386, 7.26927111972768690986858989426, 7.31280066535498996112985725664, 7.80019523012963931903075954936

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