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Properties

Label 4-1850e2-1.1-c1e2-0-7
Degree $4$
Conductor $3422500$
Sign $1$
Analytic cond. $218.221$
Root an. cond. $3.84347$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s + 3·4-s − 2·6-s + 2·7-s − 4·8-s − 4·9-s − 5·11-s + 3·12-s − 13-s − 4·14-s + 5·16-s + 8·18-s + 2·21-s + 10·22-s + 23-s − 4·24-s + 2·26-s − 6·27-s + 6·28-s − 3·29-s + 17·31-s − 6·32-s − 5·33-s − 12·36-s + 2·37-s − 39-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 3/2·4-s − 0.816·6-s + 0.755·7-s − 1.41·8-s − 4/3·9-s − 1.50·11-s + 0.866·12-s − 0.277·13-s − 1.06·14-s + 5/4·16-s + 1.88·18-s + 0.436·21-s + 2.13·22-s + 0.208·23-s − 0.816·24-s + 0.392·26-s − 1.15·27-s + 1.13·28-s − 0.557·29-s + 3.05·31-s − 1.06·32-s − 0.870·33-s − 2·36-s + 0.328·37-s − 0.160·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3422500\)    =    \(2^{2} \cdot 5^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(218.221\)
Root analytic conductor: \(3.84347\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3422500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.067839036\)
\(L(\frac12)\) \(\approx\) \(1.067839036\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
37$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_f
7$D_{4}$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_k
11$D_{4}$ \( 1 + 5 T + 27 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.11.f_bb
13$D_{4}$ \( 1 + T + 15 T^{2} + p T^{3} + p^{2} T^{4} \) 2.13.b_p
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.17.a_o
19$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.19.a_s
23$D_{4}$ \( 1 - T + 35 T^{2} - p T^{3} + p^{2} T^{4} \) 2.23.ab_bj
29$C_4$ \( 1 + 3 T - T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.29.d_ab
31$D_{4}$ \( 1 - 17 T + 133 T^{2} - 17 p T^{3} + p^{2} T^{4} \) 2.31.ar_fd
41$D_{4}$ \( 1 - 17 T + 153 T^{2} - 17 p T^{3} + p^{2} T^{4} \) 2.41.ar_fx
43$D_{4}$ \( 1 - 6 T + 90 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.43.ag_dm
47$D_{4}$ \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.47.c_dm
53$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_dy
59$D_{4}$ \( 1 + 14 T + 162 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.59.o_gg
61$D_{4}$ \( 1 - 19 T + 211 T^{2} - 19 p T^{3} + p^{2} T^{4} \) 2.61.at_id
67$D_{4}$ \( 1 - 9 T + 123 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.67.aj_et
71$D_{4}$ \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.71.m_du
73$D_{4}$ \( 1 + 3 T + 117 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.73.d_en
79$D_{4}$ \( 1 - 3 T + 59 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.79.ad_ch
83$D_{4}$ \( 1 - 20 T + 246 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.83.au_jm
89$C_4$ \( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_hm
97$D_{4}$ \( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.97.i_hi
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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

−9.207007149496274768084771497279, −9.163989806725291634246632017566, −8.460047583285389607433357308928, −8.345416302891367089728665455621, −7.85083522820591253615344971463, −7.83845619818879868059911359685, −7.43754249445562827816109448725, −6.77693211367059718523087150475, −6.30342451127997648623734366645, −5.98503522449555619853665348234, −5.36115706059252588801950825631, −5.23738465172214104059602453053, −4.47617856205482839765141515619, −4.07417128609313017282858734365, −3.12247841847445430566933046554, −2.84926902413657059545312335432, −2.42693996546770832138538433496, −2.16233851146825294382014338910, −1.11681020921238536563500437236, −0.51467502370781099341159803573, 0.51467502370781099341159803573, 1.11681020921238536563500437236, 2.16233851146825294382014338910, 2.42693996546770832138538433496, 2.84926902413657059545312335432, 3.12247841847445430566933046554, 4.07417128609313017282858734365, 4.47617856205482839765141515619, 5.23738465172214104059602453053, 5.36115706059252588801950825631, 5.98503522449555619853665348234, 6.30342451127997648623734366645, 6.77693211367059718523087150475, 7.43754249445562827816109448725, 7.83845619818879868059911359685, 7.85083522820591253615344971463, 8.345416302891367089728665455621, 8.460047583285389607433357308928, 9.163989806725291634246632017566, 9.207007149496274768084771497279

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