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Properties

Label 4-960e2-1.1-c1e2-0-27
Degree $4$
Conductor $921600$
Sign $1$
Analytic cond. $58.7620$
Root an. cond. $2.76868$
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  + 2·5-s − 9-s + 4·13-s + 4·17-s + 3·25-s − 4·29-s + 12·37-s − 4·41-s − 2·45-s + 6·49-s + 20·53-s + 4·61-s + 8·65-s − 4·73-s + 81-s + 8·85-s − 12·89-s + 12·97-s + 12·101-s − 12·109-s − 4·113-s − 4·117-s − 6·121-s + 4·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.894·5-s − 1/3·9-s + 1.10·13-s + 0.970·17-s + 3/5·25-s − 0.742·29-s + 1.97·37-s − 0.624·41-s − 0.298·45-s + 6/7·49-s + 2.74·53-s + 0.512·61-s + 0.992·65-s − 0.468·73-s + 1/9·81-s + 0.867·85-s − 1.27·89-s + 1.21·97-s + 1.19·101-s − 1.14·109-s − 0.376·113-s − 0.369·117-s − 0.545·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(921600\)    =    \(2^{12} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(58.7620\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 921600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.803917852\)
\(L(\frac12)\) \(\approx\) \(2.803917852\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
good7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.7.a_ag
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.ae_w
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.a_aba
23$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.23.a_ag
29$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.e_bu
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.31.a_be
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.37.am_dq
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.e_w
43$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.43.a_k
47$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.47.a_ag
53$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.53.au_hi
59$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.59.a_g
61$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.ae_as
67$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.67.a_ady
71$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.71.a_be
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.e_g
79$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.79.a_da
83$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \) 2.83.a_adi
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.97.am_ig
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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

−8.197583128587011294573660454990, −7.75607362005887971566007526051, −7.28990729415394696222055218269, −6.80382813403846473025826771591, −6.29005983602675476263053442911, −5.85154189950553561868529735768, −5.60078284840423417456855955663, −5.18560884013015285113930661691, −4.47077099251116239108779577658, −3.86850844409555418989432301855, −3.53113048928162608318301357137, −2.71269144652642031556550384593, −2.35014419199254126013460628200, −1.47110145224388845847332462272, −0.855985849557055733700832379488, 0.855985849557055733700832379488, 1.47110145224388845847332462272, 2.35014419199254126013460628200, 2.71269144652642031556550384593, 3.53113048928162608318301357137, 3.86850844409555418989432301855, 4.47077099251116239108779577658, 5.18560884013015285113930661691, 5.60078284840423417456855955663, 5.85154189950553561868529735768, 6.29005983602675476263053442911, 6.80382813403846473025826771591, 7.28990729415394696222055218269, 7.75607362005887971566007526051, 8.197583128587011294573660454990

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