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Properties

Label 4-1850e2-1.1-c1e2-0-13
Degree $4$
Conductor $3422500$
Sign $1$
Analytic cond. $218.221$
Root an. cond. $3.84347$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s + 4·8-s + 6·9-s + 6·11-s − 4·13-s + 5·16-s − 2·17-s + 12·18-s + 12·22-s − 12·23-s − 8·26-s + 6·32-s − 4·34-s + 18·36-s − 12·37-s + 10·41-s + 22·43-s + 18·44-s − 24·46-s − 11·49-s − 12·52-s + 7·64-s − 6·68-s + 4·71-s + 24·72-s − 24·74-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s + 2·9-s + 1.80·11-s − 1.10·13-s + 5/4·16-s − 0.485·17-s + 2.82·18-s + 2.55·22-s − 2.50·23-s − 1.56·26-s + 1.06·32-s − 0.685·34-s + 3·36-s − 1.97·37-s + 1.56·41-s + 3.35·43-s + 2.71·44-s − 3.53·46-s − 1.57·49-s − 1.66·52-s + 7/8·64-s − 0.727·68-s + 0.474·71-s + 2.82·72-s − 2.78·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3422500\)    =    \(2^{2} \cdot 5^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(218.221\)
Root analytic conductor: \(3.84347\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3422500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.177980982\)
\(L(\frac12)\) \(\approx\) \(8.177980982\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
37$C_2$ \( 1 + 12 T + p T^{2} \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.3.a_ag
7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.7.a_l
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.11.ag_bf
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.17.c_bj
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.19.a_abi
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.23.m_de
29$C_2^2$ \( 1 - 33 T^{2} + p^{2} T^{4} \) 2.29.a_abh
31$C_2^2$ \( 1 - 61 T^{2} + p^{2} T^{4} \) 2.31.a_acj
41$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.41.ak_ed
43$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.43.aw_hz
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.47.a_abe
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.53.a_az
59$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.59.a_ba
61$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \) 2.61.a_acv
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.67.a_afa
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.71.ae_fq
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.a_aeg
79$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.79.a_agc
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \) 2.89.a_agg
97$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.97.w_md
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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

−9.365433901307916757036876238777, −9.309599800347654860591978449518, −8.713008769756002695167083309848, −7.947640534138135920507473756675, −7.75049846147459723594163771463, −7.30877147646708006609159942951, −6.83921422411362489763174019776, −6.81318215614629652702502073436, −6.07253757774285045766794578071, −5.96676924548532722673408752191, −5.41401325149031698123111522902, −4.53867238438763079385203332593, −4.50061658424296055028328432439, −4.27798398581720294187128543463, −3.62683743529594789690438247827, −3.49614269472130565669986720457, −2.45054938670335094397977526086, −2.00602650260678251321626644522, −1.69417294331642415473009116422, −0.851006944572910665417832137267, 0.851006944572910665417832137267, 1.69417294331642415473009116422, 2.00602650260678251321626644522, 2.45054938670335094397977526086, 3.49614269472130565669986720457, 3.62683743529594789690438247827, 4.27798398581720294187128543463, 4.50061658424296055028328432439, 4.53867238438763079385203332593, 5.41401325149031698123111522902, 5.96676924548532722673408752191, 6.07253757774285045766794578071, 6.81318215614629652702502073436, 6.83921422411362489763174019776, 7.30877147646708006609159942951, 7.75049846147459723594163771463, 7.947640534138135920507473756675, 8.713008769756002695167083309848, 9.309599800347654860591978449518, 9.365433901307916757036876238777

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