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Properties

Label 4-5989500-1.1-c1e2-0-61
Degree $4$
Conductor $5989500$
Sign $-1$
Analytic cond. $381.895$
Root an. cond. $4.42065$
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 + 11-s + 16-s + 20-s − 8·23-s + 25-s − 8·31-s + 36-s + 8·37-s + 44-s + 45-s + 8·47-s − 6·49-s + 8·53-s + 55-s + 64-s + 16·67-s − 16·71-s + 80-s + 81-s − 4·89-s − 8·92-s − 32·97-s + 99-s + 100-s + ⋯
L(s)  = 1  + 1/2·4-s + 0.447·5-s + 1/3·9-s + 0.301·11-s + 1/4·16-s + 0.223·20-s − 1.66·23-s + 1/5·25-s − 1.43·31-s + 1/6·36-s + 1.31·37-s + 0.150·44-s + 0.149·45-s + 1.16·47-s − 6/7·49-s + 1.09·53-s + 0.134·55-s + 1/8·64-s + 1.95·67-s − 1.89·71-s + 0.111·80-s + 1/9·81-s − 0.423·89-s − 0.834·92-s − 3.24·97-s + 0.100·99-s + 1/10·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5989500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5989500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5989500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{3} \cdot 11^{3}\)
Sign: $-1$
Analytic conductor: \(381.895\)
Root analytic conductor: \(4.42065\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 5989500,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$ \( 1 - T \)
11$C_1$ \( 1 - T \)
good7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.17.a_o
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.19.a_g
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.23.i_ck
29$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.29.a_be
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.i_ck
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ai_cc
41$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.41.a_ak
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.43.a_o
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.47.ai_bu
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.53.ai_eo
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.61.a_aec
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.67.aq_hq
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.71.q_fm
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.a_bu
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.79.a_be
83$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.83.a_acg
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.e_bm
97$C_2$$\times$$C_2$ \( ( 1 + 14 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.bg_re
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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

−7.02764047786468893410061159524, −6.69267633781013070705798761515, −6.12074794069477526714514734153, −5.91961812321784426317642770274, −5.49576710021470778026187255024, −5.14879288917095349616097436684, −4.35683989133678359819980332606, −4.18742320705109712116698866506, −3.70472791311396595039713507536, −3.19004506367461730252129162130, −2.47151900154141523044882188691, −2.24708515278452201128469510182, −1.59578679051224544878361778261, −1.07427271147761036769336701657, 0, 1.07427271147761036769336701657, 1.59578679051224544878361778261, 2.24708515278452201128469510182, 2.47151900154141523044882188691, 3.19004506367461730252129162130, 3.70472791311396595039713507536, 4.18742320705109712116698866506, 4.35683989133678359819980332606, 5.14879288917095349616097436684, 5.49576710021470778026187255024, 5.91961812321784426317642770274, 6.12074794069477526714514734153, 6.69267633781013070705798761515, 7.02764047786468893410061159524

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