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Properties

Label 4-42e4-1.1-c1e2-0-21
Degree $4$
Conductor $3111696$
Sign $1$
Analytic cond. $198.404$
Root an. cond. $3.75308$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 12·13-s + 4·16-s − 8·25-s − 4·37-s + 24·52-s − 24·61-s − 8·64-s + 12·73-s + 36·97-s + 16·100-s − 40·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 8·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 82·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 4-s − 3.32·13-s + 16-s − 8/5·25-s − 0.657·37-s + 3.32·52-s − 3.07·61-s − 64-s + 1.40·73-s + 3.65·97-s + 8/5·100-s − 3.83·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.657·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3111696\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(198.404\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3111696,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + p T^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.5.a_i
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.13.m_ck
17$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.17.a_aq
19$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.19.a_abm
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.29.a_bo
31$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.31.a_ack
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \) 2.41.a_dc
43$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.43.a_adi
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2^2$ \( 1 - 56 T^{2} + p^{2} T^{4} \) 2.53.a_ace
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.61.y_kg
67$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.67.a_afe
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.79.a_agc
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 - 160 T^{2} + p^{2} T^{4} \) 2.89.a_age
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \) 2.97.abk_ty
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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

−9.349161652881765353809822967736, −8.913473278837376513964614654924, −8.159673343903772828710388348701, −7.84973090458093672194499864987, −7.61753163750030095187635270562, −7.31911151413576957600433755181, −6.75006846171928101427828774204, −6.29404491527872991979833421570, −5.74664775871706001836211185351, −5.23343056491035388589562228605, −5.06439091809005784515712229700, −4.51913885613216576765549412316, −4.32608243065745671616221940583, −3.58189478830704227084674970027, −3.18438762221193426754062370358, −2.40720020561267075692020766537, −2.19168084742163858956938977243, −1.29658426870493598493239821597, 0, 0, 1.29658426870493598493239821597, 2.19168084742163858956938977243, 2.40720020561267075692020766537, 3.18438762221193426754062370358, 3.58189478830704227084674970027, 4.32608243065745671616221940583, 4.51913885613216576765549412316, 5.06439091809005784515712229700, 5.23343056491035388589562228605, 5.74664775871706001836211185351, 6.29404491527872991979833421570, 6.75006846171928101427828774204, 7.31911151413576957600433755181, 7.61753163750030095187635270562, 7.84973090458093672194499864987, 8.159673343903772828710388348701, 8.913473278837376513964614654924, 9.349161652881765353809822967736

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