Properties

Label 4-980e2-1.1-c1e2-0-19
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s + 2·9-s + 6·11-s + 16-s − 2·18-s − 6·22-s + 8·23-s + 25-s − 32-s + 2·36-s + 4·37-s + 8·43-s + 6·44-s − 8·46-s − 50-s − 4·53-s + 64-s + 20·67-s + 20·71-s − 2·72-s − 4·74-s + 4·79-s − 5·81-s − 8·86-s − 6·88-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s + 2/3·9-s + 1.80·11-s + 1/4·16-s − 0.471·18-s − 1.27·22-s + 1.66·23-s + 1/5·25-s − 0.176·32-s + 1/3·36-s + 0.657·37-s + 1.21·43-s + 0.904·44-s − 1.17·46-s − 0.141·50-s − 0.549·53-s + 1/8·64-s + 2.44·67-s + 2.37·71-s − 0.235·72-s − 0.464·74-s + 0.450·79-s − 5/9·81-s − 0.862·86-s − 0.639·88-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: \(0\)
Selberg data: \((4,\ 960400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.082141668\)
\(L(\frac12)\) \(\approx\) \(2.082141668\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
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$ \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \)
13$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} ) \)
41$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 37 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
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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

−8.027366975273020026400887777178, −7.83793627640799082548581252165, −7.20879250232265376250088182274, −6.77820230933455951971846405361, −6.57782809010391781286002901101, −6.20107228903781990964333981045, −5.37292342895458086853336337282, −5.09700154424971948940316889393, −4.33293073680155005762941290420, −3.94640754065108654172353304781, −3.47494106061143408855061911123, −2.74429433532495952193708017562, −2.12208261364698785145198626275, −1.26440163312596409583792309937, −0.904606159115227586659809900858, 0.904606159115227586659809900858, 1.26440163312596409583792309937, 2.12208261364698785145198626275, 2.74429433532495952193708017562, 3.47494106061143408855061911123, 3.94640754065108654172353304781, 4.33293073680155005762941290420, 5.09700154424971948940316889393, 5.37292342895458086853336337282, 6.20107228903781990964333981045, 6.57782809010391781286002901101, 6.77820230933455951971846405361, 7.20879250232265376250088182274, 7.83793627640799082548581252165, 8.027366975273020026400887777178

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