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Properties

Label 4-2323200-1.1-c1e2-0-32
Degree $4$
Conductor $2323200$
Sign $1$
Analytic cond. $148.129$
Root an. cond. $3.48867$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 5-s + 4·9-s + 6·11-s − 2·15-s − 4·23-s − 4·25-s + 5·27-s + 7·31-s + 12·33-s − 6·37-s − 4·45-s + 10·47-s − 6·49-s + 5·53-s − 6·55-s + 10·59-s + 23·67-s − 8·69-s + 17·71-s − 8·75-s + 4·81-s − 23·89-s + 14·93-s − 32·97-s + 24·99-s + 103-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s + 4/3·9-s + 1.80·11-s − 0.516·15-s − 0.834·23-s − 4/5·25-s + 0.962·27-s + 1.25·31-s + 2.08·33-s − 0.986·37-s − 0.596·45-s + 1.45·47-s − 6/7·49-s + 0.686·53-s − 0.809·55-s + 1.30·59-s + 2.80·67-s − 0.963·69-s + 2.01·71-s − 0.923·75-s + 4/9·81-s − 2.43·89-s + 1.45·93-s − 3.24·97-s + 2.41·99-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2323200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2323200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2323200\)    =    \(2^{8} \cdot 3 \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(148.129\)
Root analytic conductor: \(3.48867\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2323200,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.996321364\)
\(L(\frac12)\) \(\approx\) \(3.996321364\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - p T + p T^{2} ) \)
5$C_2$ \( 1 + T + p T^{2} \)
11$C_2$ \( 1 - 6 T + p T^{2} \)
good7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.13.a_b
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.17.a_q
19$C_2^2$ \( 1 - 24 T^{2} + p^{2} T^{4} \) 2.19.a_ay
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.e_bu
29$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.29.a_abe
31$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.31.ah_bg
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.37.g_de
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.a_ack
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.43.a_aby
47$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.47.ak_el
53$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.af_cs
59$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.59.ak_fm
61$C_2^2$ \( 1 - 105 T^{2} + p^{2} T^{4} \) 2.61.a_aeb
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 11 T + p T^{2} ) \) 2.67.ax_kg
71$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.71.ar_ig
73$C_2^2$ \( 1 - 125 T^{2} + p^{2} T^{4} \) 2.73.a_aev
79$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.79.a_bi
83$C_2^2$ \( 1 + 103 T^{2} + p^{2} T^{4} \) 2.83.a_dz
89$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.89.x_lm
97$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.97.bg_ri
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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

−7.915061478363290872661758998005, −7.11786298087862154737252973066, −6.90521902753137301839242866842, −6.69065782753063263465820590021, −6.09987253028993883733146008575, −5.50975372268923174319813342746, −5.10515732632393841360693043960, −4.21640035302465590805986938535, −4.04290603700301575320151155966, −3.89925168053493854092457981887, −3.26509435062143808420539812403, −2.61457920457380198374674187575, −2.06050680887402525258822597405, −1.50902200900300362267203053971, −0.77880001681457936323746867989, 0.77880001681457936323746867989, 1.50902200900300362267203053971, 2.06050680887402525258822597405, 2.61457920457380198374674187575, 3.26509435062143808420539812403, 3.89925168053493854092457981887, 4.04290603700301575320151155966, 4.21640035302465590805986938535, 5.10515732632393841360693043960, 5.50975372268923174319813342746, 6.09987253028993883733146008575, 6.69065782753063263465820590021, 6.90521902753137301839242866842, 7.11786298087862154737252973066, 7.915061478363290872661758998005

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