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Properties

Label 4-1782e2-1.1-c1e2-0-28
Degree $4$
Conductor $3175524$
Sign $1$
Analytic cond. $202.474$
Root an. cond. $3.77217$
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  + 4-s − 6·5-s + 16-s − 6·20-s + 17·25-s − 8·31-s − 2·37-s − 24·47-s + 2·49-s − 12·53-s + 64-s − 8·67-s − 24·71-s − 6·80-s − 6·89-s + 4·97-s + 17·100-s − 8·103-s − 30·113-s − 11·121-s − 8·124-s − 18·125-s + 127-s + 131-s + 137-s + 139-s − 2·148-s + ⋯
L(s)  = 1  + 1/2·4-s − 2.68·5-s + 1/4·16-s − 1.34·20-s + 17/5·25-s − 1.43·31-s − 0.328·37-s − 3.50·47-s + 2/7·49-s − 1.64·53-s + 1/8·64-s − 0.977·67-s − 2.84·71-s − 0.670·80-s − 0.635·89-s + 0.406·97-s + 1.69·100-s − 0.788·103-s − 2.82·113-s − 121-s − 0.718·124-s − 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.164·148-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3175524 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3175524 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3175524\)    =    \(2^{2} \cdot 3^{8} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(202.474\)
Root analytic conductor: \(3.77217\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3175524,\ (\ :1/2, 1/2),\ 1)\)

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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3 \( 1 \)
11$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.5.g_t
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.13.a_z
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.a_z
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.29.a_ax
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.37.c_cx
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.47.y_je
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.61.a_er
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.73.a_z
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.a_adu
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.89.g_hf
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

−7.16277862972928710155421440227, −6.98504291328603819496217788978, −6.29261335708839240986736404399, −6.07141567781617112259653788579, −5.16414682280697373873566197697, −5.04773425377657589958068791936, −4.25638665319375785757127060430, −4.17461543307518243917716468023, −3.57784119065864589215915440592, −3.12147196896030897278593721655, −2.94651413139897301734292877776, −1.84842910920498472471311986628, −1.34591687981300633859944824892, 0, 0, 1.34591687981300633859944824892, 1.84842910920498472471311986628, 2.94651413139897301734292877776, 3.12147196896030897278593721655, 3.57784119065864589215915440592, 4.17461543307518243917716468023, 4.25638665319375785757127060430, 5.04773425377657589958068791936, 5.16414682280697373873566197697, 6.07141567781617112259653788579, 6.29261335708839240986736404399, 6.98504291328603819496217788978, 7.16277862972928710155421440227

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