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Properties

Label 4-540800-1.1-c1e2-0-66
Degree $4$
Conductor $540800$
Sign $-1$
Analytic cond. $34.4818$
Root an. cond. $2.42324$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 9-s − 2·13-s + 2·17-s − 25-s − 29-s − 4·37-s − 6·41-s − 2·45-s − 9·49-s − 19·53-s + 61-s − 4·65-s + 6·73-s − 8·81-s + 4·85-s − 26·89-s − 2·97-s − 5·101-s + 14·109-s + 26·113-s + 2·117-s + 7·121-s − 12·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.894·5-s − 1/3·9-s − 0.554·13-s + 0.485·17-s − 1/5·25-s − 0.185·29-s − 0.657·37-s − 0.937·41-s − 0.298·45-s − 9/7·49-s − 2.60·53-s + 0.128·61-s − 0.496·65-s + 0.702·73-s − 8/9·81-s + 0.433·85-s − 2.75·89-s − 0.203·97-s − 0.497·101-s + 1.34·109-s + 2.44·113-s + 0.184·117-s + 7/11·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(540800\)    =    \(2^{7} \cdot 5^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(34.4818\)
Root analytic conductor: \(2.42324\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 540800,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
7$C_2^2$ \( 1 + 9 T^{2} + p^{2} T^{4} \) 2.7.a_j
11$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \) 2.11.a_ah
17$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.17.ac_bf
19$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.19.a_ao
23$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \) 2.23.a_al
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.29.b_q
31$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.31.a_z
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.g_bq
43$C_2^2$ \( 1 + 41 T^{2} + p^{2} T^{4} \) 2.43.a_bp
47$C_2^2$ \( 1 + 77 T^{2} + p^{2} T^{4} \) 2.47.a_cz
53$C_2$$\times$$C_2$ \( ( 1 + 9 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.t_ho
59$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.59.a_z
61$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.ab_dc
67$C_2^2$ \( 1 - 87 T^{2} + p^{2} T^{4} \) 2.67.a_adj
71$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \) 2.71.a_adu
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.73.ag_bi
79$C_2^2$ \( 1 - 71 T^{2} + p^{2} T^{4} \) 2.79.a_act
83$C_2^2$ \( 1 - 15 T^{2} + p^{2} T^{4} \) 2.83.a_ap
89$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.89.ba_na
97$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.c_ba
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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

−8.225457011769635127603366987366, −7.902360070672120744838521187489, −7.24931711813408962529506824590, −6.92366460218482982346568505974, −6.30005929447839619762995140764, −5.97707206742271818223619134933, −5.48540942993696856255618086177, −4.95752653635396269834516611094, −4.63527967374116732471529633381, −3.79304489941910565947003882615, −3.23465046512312398116175487026, −2.72644796158420944393983593719, −1.93254420001541197697437762494, −1.44070204222194142618697579148, 0, 1.44070204222194142618697579148, 1.93254420001541197697437762494, 2.72644796158420944393983593719, 3.23465046512312398116175487026, 3.79304489941910565947003882615, 4.63527967374116732471529633381, 4.95752653635396269834516611094, 5.48540942993696856255618086177, 5.97707206742271818223619134933, 6.30005929447839619762995140764, 6.92366460218482982346568505974, 7.24931711813408962529506824590, 7.902360070672120744838521187489, 8.225457011769635127603366987366

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