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Properties

Label 4-64900-1.1-c1e2-0-11
Degree $4$
Conductor $64900$
Sign $-1$
Analytic cond. $4.13808$
Root an. cond. $1.42626$
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  − 4-s + 5-s − 9-s − 4·11-s + 16-s − 5·19-s − 20-s − 4·25-s − 3·29-s − 9·31-s + 36-s − 9·41-s + 4·44-s − 45-s − 4·49-s − 4·55-s + 8·59-s + 6·61-s − 64-s + 6·71-s + 5·76-s + 17·79-s + 80-s − 8·81-s − 6·89-s − 5·95-s + 4·99-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.447·5-s − 1/3·9-s − 1.20·11-s + 1/4·16-s − 1.14·19-s − 0.223·20-s − 4/5·25-s − 0.557·29-s − 1.61·31-s + 1/6·36-s − 1.40·41-s + 0.603·44-s − 0.149·45-s − 4/7·49-s − 0.539·55-s + 1.04·59-s + 0.768·61-s − 1/8·64-s + 0.712·71-s + 0.573·76-s + 1.91·79-s + 0.111·80-s − 8/9·81-s − 0.635·89-s − 0.512·95-s + 0.402·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(64900\)    =    \(2^{2} \cdot 5^{2} \cdot 11 \cdot 59\)
Sign: $-1$
Analytic conductor: \(4.13808\)
Root analytic conductor: \(1.42626\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 64900,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - T + p T^{2} \)
11$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 3 T + p T^{2} ) \)
59$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 7 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
13$C_2^2$ \( 1 + 21 T^{2} + p^{2} T^{4} \) 2.13.a_v
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.17.a_ao
19$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.f_bg
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.29.d_cg
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.31.j_ck
37$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.37.a_u
41$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.j_bu
43$C_2^2$ \( 1 - 39 T^{2} + p^{2} T^{4} \) 2.43.a_abn
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.47.a_da
53$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \) 2.53.a_bf
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.ag_ec
67$C_2^2$ \( 1 + 112 T^{2} + p^{2} T^{4} \) 2.67.a_ei
71$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.71.ag_bz
73$C_2^2$ \( 1 - 27 T^{2} + p^{2} T^{4} \) 2.73.a_abb
79$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.79.ar_iw
83$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.83.a_g
89$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.89.g_gg
97$C_2^2$ \( 1 + 120 T^{2} + p^{2} T^{4} \) 2.97.a_eq
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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

−9.751713028675597506440413564900, −9.049565732689380267605823364395, −8.733294790824389049138288078310, −8.066110993575480013835495004115, −7.80434066807409705631082291636, −7.04060110426205116364974565436, −6.49437658724063059601167271369, −5.80348200434042310293811681685, −5.32536020408280156342123520676, −4.95333295052437807209024255245, −4.01399087529480276562748018740, −3.51237668815602679081221418320, −2.51117174047432405576660443261, −1.85198340641021533196213602399, 0, 1.85198340641021533196213602399, 2.51117174047432405576660443261, 3.51237668815602679081221418320, 4.01399087529480276562748018740, 4.95333295052437807209024255245, 5.32536020408280156342123520676, 5.80348200434042310293811681685, 6.49437658724063059601167271369, 7.04060110426205116364974565436, 7.80434066807409705631082291636, 8.066110993575480013835495004115, 8.733294790824389049138288078310, 9.049565732689380267605823364395, 9.751713028675597506440413564900

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