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Properties

Label 4-100352-1.1-c1e2-0-4
Degree $4$
Conductor $100352$
Sign $1$
Analytic cond. $6.39853$
Root an. cond. $1.59045$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·9-s + 8·23-s − 2·25-s + 16·31-s − 8·41-s − 7·49-s + 8·71-s + 24·73-s − 8·79-s − 5·81-s + 24·89-s + 8·103-s − 4·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 2/3·9-s + 1.66·23-s − 2/5·25-s + 2.87·31-s − 1.24·41-s − 49-s + 0.949·71-s + 2.80·73-s − 0.900·79-s − 5/9·81-s + 2.54·89-s + 0.788·103-s − 0.376·113-s − 0.545·121-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 + 0.0783·163-s + 0.0773·167-s − 0.153·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100352 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100352 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(100352\)    =    \(2^{11} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(6.39853\)
Root analytic conductor: \(1.59045\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 100352,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.444043622\)
\(L(\frac12)\) \(\approx\) \(1.444043622\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.13.a_c
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.19.a_s
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.23.ai_bu
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.29.a_bm
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.31.aq_ew
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.i_dq
43$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.43.a_aba
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.53.a_acg
59$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.59.a_ao
61$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \) 2.61.a_aeg
67$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.67.a_acg
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.ai_fm
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) 2.73.ay_la
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.i_be
83$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \) 2.83.a_ek
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) 2.89.ay_mg
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.a_ac
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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

−9.545940243699197452112309026094, −9.072366720648179752841300803401, −8.545436371357290455893090503111, −8.118887716777139084071953049362, −7.75085107038760568990576253414, −6.90311089622575463992498908658, −6.56049674856934005097609395622, −6.12603238503703836674830996619, −5.26799158514649832689175561102, −4.96118990897796589243612936068, −4.33002557609838729383661226785, −3.40190696506899088672966011590, −2.96613495701333264850912486308, −2.14828327896835123257603408212, −0.915315312961161516616311421919, 0.915315312961161516616311421919, 2.14828327896835123257603408212, 2.96613495701333264850912486308, 3.40190696506899088672966011590, 4.33002557609838729383661226785, 4.96118990897796589243612936068, 5.26799158514649832689175561102, 6.12603238503703836674830996619, 6.56049674856934005097609395622, 6.90311089622575463992498908658, 7.75085107038760568990576253414, 8.118887716777139084071953049362, 8.545436371357290455893090503111, 9.072366720648179752841300803401, 9.545940243699197452112309026094

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