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Properties

Label 4-440e2-1.1-c1e2-0-13
Degree $4$
Conductor $193600$
Sign $-1$
Analytic cond. $12.3441$
Root an. cond. $1.87441$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 4·7-s − 2·9-s + 8·19-s + 3·25-s + 8·35-s + 4·37-s + 20·43-s + 4·45-s − 2·49-s − 12·53-s + 8·63-s − 16·79-s − 5·81-s − 12·83-s − 12·89-s − 16·95-s + 4·97-s + 12·107-s − 12·113-s − 11·121-s − 4·125-s + 127-s + 131-s − 32·133-s + 137-s + 139-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.51·7-s − 2/3·9-s + 1.83·19-s + 3/5·25-s + 1.35·35-s + 0.657·37-s + 3.04·43-s + 0.596·45-s − 2/7·49-s − 1.64·53-s + 1.00·63-s − 1.80·79-s − 5/9·81-s − 1.31·83-s − 1.27·89-s − 1.64·95-s + 0.406·97-s + 1.16·107-s − 1.12·113-s − 121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s − 2.77·133-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(193600\)    =    \(2^{6} \cdot 5^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(12.3441\)
Root analytic conductor: \(1.87441\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 193600,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( ( 1 + T )^{2} \)
11$C_2$ \( 1 + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.43.au_he
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.a_eo
67$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.67.a_fa
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.83.m_hu
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−8.740293324354028093320034492248, −8.597075432672291247955228431981, −7.71273110823499542846308181544, −7.48339952967373098458922175110, −7.12025897518162428207793885558, −6.27087624192875571051851265258, −6.09088770920212572077792597538, −5.46752609716119299096669229449, −4.83331995302090139893539895632, −4.12250433368686324236236368171, −3.61120105048083329633068803983, −2.90312545354050643736888796120, −2.76929890617261215013507568311, −1.17768193714929585417069498904, 0, 1.17768193714929585417069498904, 2.76929890617261215013507568311, 2.90312545354050643736888796120, 3.61120105048083329633068803983, 4.12250433368686324236236368171, 4.83331995302090139893539895632, 5.46752609716119299096669229449, 6.09088770920212572077792597538, 6.27087624192875571051851265258, 7.12025897518162428207793885558, 7.48339952967373098458922175110, 7.71273110823499542846308181544, 8.597075432672291247955228431981, 8.740293324354028093320034492248

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