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Properties

Label 4-1350e2-1.1-c1e2-0-22
Degree $4$
Conductor $1822500$
Sign $1$
Analytic cond. $116.204$
Root an. cond. $3.28326$
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-s − 4·7-s − 8-s + 3·11-s − 4·13-s − 4·14-s − 16-s − 6·17-s − 8·19-s + 3·22-s + 6·23-s − 4·26-s − 6·29-s − 8·31-s − 6·34-s − 16·37-s − 8·38-s − 6·41-s − 43-s + 6·46-s − 12·47-s + 7·49-s + 4·56-s − 6·58-s − 9·59-s − 8·61-s − 8·62-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.51·7-s − 0.353·8-s + 0.904·11-s − 1.10·13-s − 1.06·14-s − 1/4·16-s − 1.45·17-s − 1.83·19-s + 0.639·22-s + 1.25·23-s − 0.784·26-s − 1.11·29-s − 1.43·31-s − 1.02·34-s − 2.63·37-s − 1.29·38-s − 0.937·41-s − 0.152·43-s + 0.884·46-s − 1.75·47-s + 49-s + 0.534·56-s − 0.787·58-s − 1.17·59-s − 1.02·61-s − 1.01·62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1822500\)    =    \(2^{2} \cdot 3^{6} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(116.204\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1822500,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
5 \( 1 \)
good7$C_2$ \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.e_j
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_ac
13$C_2^2$ \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_d
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.17.g_br
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_n
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.29.g_h
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_bh
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.37.q_fi
41$C_2^2$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_af
43$C_2^2$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) 2.43.b_abq
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.47.m_dt
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2^2$ \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.59.j_w
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_d
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_abz
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.71.am_gw
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.73.bc_ne
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_ap
83$C_2^2$ \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.83.aj_ac
89$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.89.as_jz
97$C_2^2$ \( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.97.h_abw
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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

−9.209949135907358853707789453404, −9.072565705672094122840826390806, −8.716156967957626730750474271484, −8.492830839370149759457323954936, −7.47226948155862627954468333886, −7.21479440082634321312809627087, −6.92379470558340925961329827373, −6.33940320284483676222591412271, −6.30019813578637055781990469776, −5.72249825854272382114369910028, −4.94303188473312959133888086182, −4.81967146148831469765363247224, −4.30963274915364172806476699329, −3.59923591723418374851085855663, −3.41453047061195233080945793959, −2.93649478361689139650692007268, −1.94679271358269380536154130089, −1.86973378363312815574764383229, 0, 0, 1.86973378363312815574764383229, 1.94679271358269380536154130089, 2.93649478361689139650692007268, 3.41453047061195233080945793959, 3.59923591723418374851085855663, 4.30963274915364172806476699329, 4.81967146148831469765363247224, 4.94303188473312959133888086182, 5.72249825854272382114369910028, 6.30019813578637055781990469776, 6.33940320284483676222591412271, 6.92379470558340925961329827373, 7.21479440082634321312809627087, 7.47226948155862627954468333886, 8.492830839370149759457323954936, 8.716156967957626730750474271484, 9.072565705672094122840826390806, 9.209949135907358853707789453404

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