Properties

Label 4-1400e2-1.1-c0e2-0-2
Degree $4$
Conductor $1960000$
Sign $1$
Analytic cond. $0.488169$
Root an. cond. $0.835877$
Motivic weight $0$
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  − 4-s + 2·9-s + 16-s − 2·36-s − 49-s − 64-s − 4·71-s + 4·79-s + 3·81-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 196-s + ⋯
L(s)  = 1  − 4-s + 2·9-s + 16-s − 2·36-s − 49-s − 64-s − 4·71-s + 4·79-s + 3·81-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 196-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(1960000\)    =    \(2^{6} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.488169\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1960000,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.024322428\)
\(L(\frac12)\) \(\approx\) \(1.024322428\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good3$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_1$ \( ( 1 + T )^{4} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_1$ \( ( 1 - T )^{4} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_2$ \( ( 1 + T^{2} )^{2} \)
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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

−9.765012748437688413368988412335, −9.602136215530039227366546270583, −9.273086296306183427017120347383, −8.866905105064671329983701953689, −8.278973365732278437695112623326, −8.002590247929937488962729954525, −7.55305362595740159195052512078, −7.16429588126950139333433034383, −6.81684790072022592688546797161, −6.23411258645905093824127678128, −5.87814879234559160367791851818, −5.22797957112767086184052552108, −4.83873280622959609951008350304, −4.32135749502739859126499642656, −4.27687517674649775287371903480, −3.42991126237455350429414279205, −3.24988279972240346684076149846, −2.19301970758838871135275432597, −1.62674609263186377695371728245, −0.954165853816513525998763122907, 0.954165853816513525998763122907, 1.62674609263186377695371728245, 2.19301970758838871135275432597, 3.24988279972240346684076149846, 3.42991126237455350429414279205, 4.27687517674649775287371903480, 4.32135749502739859126499642656, 4.83873280622959609951008350304, 5.22797957112767086184052552108, 5.87814879234559160367791851818, 6.23411258645905093824127678128, 6.81684790072022592688546797161, 7.16429588126950139333433034383, 7.55305362595740159195052512078, 8.002590247929937488962729954525, 8.278973365732278437695112623326, 8.866905105064671329983701953689, 9.273086296306183427017120347383, 9.602136215530039227366546270583, 9.765012748437688413368988412335

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