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Properties

Label 4-836352-1.1-c1e2-0-21
Degree $4$
Conductor $836352$
Sign $-1$
Analytic cond. $53.3265$
Root an. cond. $2.70231$
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  − 3-s − 4·5-s + 9-s − 4·11-s + 4·15-s + 6·25-s − 27-s + 4·33-s + 4·37-s − 4·45-s + 16·47-s + 2·49-s − 12·53-s + 16·55-s − 8·59-s + 8·67-s − 6·75-s + 81-s − 32·89-s + 4·97-s − 4·99-s − 16·103-s − 4·111-s + 32·113-s + 5·121-s − 4·125-s + 127-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s + 1/3·9-s − 1.20·11-s + 1.03·15-s + 6/5·25-s − 0.192·27-s + 0.696·33-s + 0.657·37-s − 0.596·45-s + 2.33·47-s + 2/7·49-s − 1.64·53-s + 2.15·55-s − 1.04·59-s + 0.977·67-s − 0.692·75-s + 1/9·81-s − 3.39·89-s + 0.406·97-s − 0.402·99-s − 1.57·103-s − 0.379·111-s + 3.01·113-s + 5/11·121-s − 0.357·125-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 836352 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 836352 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(836352\)    =    \(2^{8} \cdot 3^{3} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(53.3265\)
Root analytic conductor: \(2.70231\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 836352,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_1$ \( 1 + T \)
11$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.e_k
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.13.a_ag
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.17.a_ao
19$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.19.a_ag
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.29.a_k
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ae_ck
41$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.41.a_by
43$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.43.a_abm
47$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.47.aq_fy
53$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.53.m_fi
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.i_eo
61$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.61.a_acs
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.71.a_fi
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.a_eg
79$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.79.a_as
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.83.a_adm
89$C_2$$\times$$C_2$ \( ( 1 + 14 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.89.bg_qo
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

−7.945622016502243224847701104609, −7.63139923956999230733910515116, −7.18577634617383065439787247940, −6.84083935439291598381562154099, −6.17570038126434784043106649634, −5.65440534329059999143681488250, −5.32025359461810653460090162529, −4.64335485818663032049696284521, −4.25944894660810830568687341884, −3.94809617382619313130256729441, −3.15351027376708913083788529260, −2.79748673584511638371622101119, −1.90493554968336076662460676170, −0.796514085123778837584970571140, 0, 0.796514085123778837584970571140, 1.90493554968336076662460676170, 2.79748673584511638371622101119, 3.15351027376708913083788529260, 3.94809617382619313130256729441, 4.25944894660810830568687341884, 4.64335485818663032049696284521, 5.32025359461810653460090162529, 5.65440534329059999143681488250, 6.17570038126434784043106649634, 6.84083935439291598381562154099, 7.18577634617383065439787247940, 7.63139923956999230733910515116, 7.945622016502243224847701104609

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