Properties

Label 4-980e2-1.1-c1e2-0-27
Degree $4$
Conductor $960400$
Sign $-1$
Analytic cond. $61.2359$
Root an. cond. $2.79738$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 3·8-s + 2·9-s + 4·11-s − 16-s − 2·18-s − 4·22-s + 4·23-s + 25-s − 5·32-s − 2·36-s − 20·37-s − 4·43-s − 4·44-s − 4·46-s − 50-s + 8·53-s + 7·64-s − 8·67-s − 12·71-s + 6·72-s + 20·74-s + 4·79-s − 5·81-s + 4·86-s + 12·88-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.06·8-s + 2/3·9-s + 1.20·11-s − 1/4·16-s − 0.471·18-s − 0.852·22-s + 0.834·23-s + 1/5·25-s − 0.883·32-s − 1/3·36-s − 3.28·37-s − 0.609·43-s − 0.603·44-s − 0.589·46-s − 0.141·50-s + 1.09·53-s + 7/8·64-s − 0.977·67-s − 1.42·71-s + 0.707·72-s + 2.32·74-s + 0.450·79-s − 5/9·81-s + 0.431·86-s + 1.27·88-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(960400\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{4}\)
Sign: $-1$
Analytic conductor: \(61.2359\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 960400,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7 \( 1 \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
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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

−8.038959738483472522625344908113, −7.30253895220535151648636726974, −7.20732777244403505922853222891, −6.82341677731459228692565952406, −6.27335381081189810983185767619, −5.65661774889156874502643200793, −5.09173380414656901481636344705, −4.77591609071825090281162346613, −4.20648989306295640420561851054, −3.66451821544217414798686071226, −3.36198027275384227529391531819, −2.37733682211854170704863740349, −1.49432146018316215927687130478, −1.26575477722673737406895852980, 0, 1.26575477722673737406895852980, 1.49432146018316215927687130478, 2.37733682211854170704863740349, 3.36198027275384227529391531819, 3.66451821544217414798686071226, 4.20648989306295640420561851054, 4.77591609071825090281162346613, 5.09173380414656901481636344705, 5.65661774889156874502643200793, 6.27335381081189810983185767619, 6.82341677731459228692565952406, 7.20732777244403505922853222891, 7.30253895220535151648636726974, 8.038959738483472522625344908113

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