Properties

Label 4-1400e2-1.1-c1e2-0-0
Degree $4$
Conductor $1960000$
Sign $1$
Analytic cond. $124.971$
Root an. cond. $3.34350$
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  − 3·9-s − 10·11-s + 4·19-s − 14·29-s + 8·31-s − 24·41-s − 49-s + 8·59-s + 8·61-s + 6·79-s + 30·99-s − 36·101-s − 10·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s − 12·171-s + 173-s + 179-s + ⋯
L(s)  = 1  − 9-s − 3.01·11-s + 0.917·19-s − 2.59·29-s + 1.43·31-s − 3.74·41-s − 1/7·49-s + 1.04·59-s + 1.02·61-s + 0.675·79-s + 3.01·99-s − 3.58·101-s − 0.957·109-s + 4.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1/13·169-s − 0.917·171-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1960000\)    =    \(2^{6} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(124.971\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1960000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2236514327\)
\(L(\frac12)\) \(\approx\) \(0.2236514327\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 93 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
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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

−9.912945338922367994741340918009, −9.454077248532162809256220197409, −8.835040812046141637034691858472, −8.458500745950601494457365975183, −8.116839195717068430291532311998, −7.83595882214809795956506854805, −7.52092873245584717882288237667, −6.85775665142860311628507074144, −6.65987578942819927692899991328, −5.81848835328526736895234409555, −5.46012397543347808758332835704, −5.15342213409149017338338020428, −5.14392213520221523780011852313, −4.26048787228535012777427976783, −3.48323871218730750841209915753, −3.25862285975149404729416265583, −2.54634699089576332556645417472, −2.36356503599207999051583964960, −1.47810162955999494268069435192, −0.18358137541973819908023094126, 0.18358137541973819908023094126, 1.47810162955999494268069435192, 2.36356503599207999051583964960, 2.54634699089576332556645417472, 3.25862285975149404729416265583, 3.48323871218730750841209915753, 4.26048787228535012777427976783, 5.14392213520221523780011852313, 5.15342213409149017338338020428, 5.46012397543347808758332835704, 5.81848835328526736895234409555, 6.65987578942819927692899991328, 6.85775665142860311628507074144, 7.52092873245584717882288237667, 7.83595882214809795956506854805, 8.116839195717068430291532311998, 8.458500745950601494457365975183, 8.835040812046141637034691858472, 9.454077248532162809256220197409, 9.912945338922367994741340918009

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