Properties

Label 4-980e2-1.1-c1e2-0-8
Degree $4$
Conductor $960400$
Sign $1$
Analytic cond. $61.2359$
Root an. cond. $2.79738$
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  + 2·2-s + 2·4-s − 2·5-s − 4·10-s + 10·13-s − 4·16-s − 10·17-s − 4·20-s − 25-s + 20·26-s − 8·32-s − 20·34-s + 14·37-s + 20·41-s − 2·50-s + 20·52-s + 18·53-s − 20·61-s − 8·64-s − 20·65-s − 20·68-s − 10·73-s + 28·74-s + 8·80-s − 9·81-s + 40·82-s + 20·85-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 0.894·5-s − 1.26·10-s + 2.77·13-s − 16-s − 2.42·17-s − 0.894·20-s − 1/5·25-s + 3.92·26-s − 1.41·32-s − 3.42·34-s + 2.30·37-s + 3.12·41-s − 0.282·50-s + 2.77·52-s + 2.47·53-s − 2.56·61-s − 64-s − 2.48·65-s − 2.42·68-s − 1.17·73-s + 3.25·74-s + 0.894·80-s − 81-s + 4.41·82-s + 2.16·85-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: induced by $\chi_{980} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 960400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.475544992\)
\(L(\frac12)\) \(\approx\) \(3.475544992\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
7 \( 1 \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 - p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + p^{2} T^{4} \)
47$C_2^2$ \( 1 + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + p^{2} T^{4} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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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

−10.67599034945331494902288757434, −9.627981487260499818513518465931, −9.330205966327137550518449663569, −8.828725030890345368320182968308, −8.562429496725344523077579243404, −8.241725848428945553568723512438, −7.38671161124681677231404147115, −7.38248269074306429111710550001, −6.45133865268278004635174474375, −6.33769607777875777369798015571, −5.83895897040164102096025854604, −5.67053290322739446642823771052, −4.62836210447680006054129101883, −4.33533192766915908418348351732, −3.98861786409896714761697870325, −3.85994512540815833098706395062, −2.86041093076571684453023707476, −2.65249025480493873713176071184, −1.70548983414259502956387981068, −0.69954960253167344242305936241, 0.69954960253167344242305936241, 1.70548983414259502956387981068, 2.65249025480493873713176071184, 2.86041093076571684453023707476, 3.85994512540815833098706395062, 3.98861786409896714761697870325, 4.33533192766915908418348351732, 4.62836210447680006054129101883, 5.67053290322739446642823771052, 5.83895897040164102096025854604, 6.33769607777875777369798015571, 6.45133865268278004635174474375, 7.38248269074306429111710550001, 7.38671161124681677231404147115, 8.241725848428945553568723512438, 8.562429496725344523077579243404, 8.828725030890345368320182968308, 9.330205966327137550518449663569, 9.627981487260499818513518465931, 10.67599034945331494902288757434

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