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Properties

Label 4-1095200-1.1-c1e2-0-3
Degree $4$
Conductor $1095200$
Sign $1$
Analytic cond. $69.8309$
Root an. cond. $2.89075$
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-s + 4-s − 8-s + 4·9-s + 2·13-s + 16-s − 6·17-s − 4·18-s + 25-s − 2·26-s + 6·29-s − 32-s + 6·34-s + 4·36-s + 10·37-s + 8·41-s + 8·49-s − 50-s + 2·52-s + 4·53-s − 6·58-s − 4·61-s + 64-s − 6·68-s − 4·72-s + 14·73-s − 10·74-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s + 4/3·9-s + 0.554·13-s + 1/4·16-s − 1.45·17-s − 0.942·18-s + 1/5·25-s − 0.392·26-s + 1.11·29-s − 0.176·32-s + 1.02·34-s + 2/3·36-s + 1.64·37-s + 1.24·41-s + 8/7·49-s − 0.141·50-s + 0.277·52-s + 0.549·53-s − 0.787·58-s − 0.512·61-s + 1/8·64-s − 0.727·68-s − 0.471·72-s + 1.63·73-s − 1.16·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1095200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1095200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1095200\)    =    \(2^{5} \cdot 5^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(69.8309\)
Root analytic conductor: \(2.89075\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1095200,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.763965990\)
\(L(\frac12)\) \(\approx\) \(1.763965990\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
37$C_2$ \( 1 - 10 T + p T^{2} \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.3.a_ae
7$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.7.a_ai
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.ac_c
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.g_bi
19$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.19.a_ao
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.23.a_o
29$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.ag_bq
31$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.31.a_acg
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.41.ai_dq
43$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.43.a_ac
47$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.47.a_i
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.ae_dq
59$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.59.a_k
61$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.e_ck
67$C_2^2$ \( 1 + 100 T^{2} + p^{2} T^{4} \) 2.67.a_dw
71$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.71.a_ada
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.73.ao_he
79$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.79.a_es
83$C_2^2$ \( 1 + 104 T^{2} + p^{2} T^{4} \) 2.83.a_ea
89$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.q_je
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.97.bc_pa
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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

−8.052700521306068355154467815525, −7.73321545421056499592818474766, −7.14402187319768748175862396884, −6.76038559340692364196352977609, −6.61788361625359473231816104581, −5.87955122844590609701218563093, −5.62302290497212828025289457609, −4.73701999652650432772590103483, −4.29093457310253980774409643650, −4.16581696021772974346531909187, −3.31819034784938646848156966615, −2.59518453351647994257156665957, −2.19267291046544709938962882663, −1.35439208845526395610118875502, −0.74519222623794018928794081655, 0.74519222623794018928794081655, 1.35439208845526395610118875502, 2.19267291046544709938962882663, 2.59518453351647994257156665957, 3.31819034784938646848156966615, 4.16581696021772974346531909187, 4.29093457310253980774409643650, 4.73701999652650432772590103483, 5.62302290497212828025289457609, 5.87955122844590609701218563093, 6.61788361625359473231816104581, 6.76038559340692364196352977609, 7.14402187319768748175862396884, 7.73321545421056499592818474766, 8.052700521306068355154467815525

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