Properties

Label 4-1400e2-1.1-c1e2-0-11
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  + 2·9-s + 4·19-s − 4·29-s + 8·31-s − 4·41-s − 49-s − 12·59-s + 8·61-s + 16·79-s − 5·81-s − 20·89-s + 24·101-s − 20·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 8·171-s + 173-s + 179-s + ⋯
L(s)  = 1  + 2/3·9-s + 0.917·19-s − 0.742·29-s + 1.43·31-s − 0.624·41-s − 1/7·49-s − 1.56·59-s + 1.02·61-s + 1.80·79-s − 5/9·81-s − 2.11·89-s + 2.38·101-s − 1.91·109-s − 2·121-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 + 2·169-s + 0.611·171-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

\(L(1)\) \(\approx\) \(2.275712104\)
\(L(\frac12)\) \(\approx\) \(2.275712104\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−9.782137192299039015420428127973, −9.325804699386767276234996236196, −9.144724936973781732491592415782, −8.397889481519338765926418803948, −8.267844583602414184539988046573, −7.60097980278152930322486802212, −7.49480057458445876057315126169, −6.89597773728039066374595571624, −6.50027241114960682451324655593, −6.19706853742635374274902779089, −5.46989344098252350302431261500, −5.25351275791154087860041031210, −4.71208934799557050061407574633, −4.18472634125489876184966099370, −3.84425622099164970912232649079, −3.10451445644428243907641366325, −2.82993948735686424815583918179, −1.96510182166407410189717775949, −1.45497257569858995093002737719, −0.64455666675166078320055723254, 0.64455666675166078320055723254, 1.45497257569858995093002737719, 1.96510182166407410189717775949, 2.82993948735686424815583918179, 3.10451445644428243907641366325, 3.84425622099164970912232649079, 4.18472634125489876184966099370, 4.71208934799557050061407574633, 5.25351275791154087860041031210, 5.46989344098252350302431261500, 6.19706853742635374274902779089, 6.50027241114960682451324655593, 6.89597773728039066374595571624, 7.49480057458445876057315126169, 7.60097980278152930322486802212, 8.267844583602414184539988046573, 8.397889481519338765926418803948, 9.144724936973781732491592415782, 9.325804699386767276234996236196, 9.782137192299039015420428127973

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