Properties

Label 4-980e2-1.1-c1e2-0-10
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

\(L(1)\) \(\approx\) \(4.468557847\)
\(L(\frac12)\) \(\approx\) \(4.468557847\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−10.07777529203296053239248113589, −9.995895005827938782847657328128, −9.630449055576491865947824382297, −9.086489278559170336108820841607, −8.523325819574670859769689729456, −7.961926639505284974667826140339, −7.54973587624073372361618498502, −7.16140182094123318922567456520, −6.73667846196512459043785934601, −6.26070345883133105806874898042, −5.54395070645542367514017902965, −5.33941418353216564672989958363, −5.26653230877033484086553748573, −4.51687053298683043767101964001, −4.11372971456978406634294044017, −3.28457437384958610690493771457, −3.07478254024297055431198054974, −2.19804625517662567576110363493, −2.11859904984605800539185865731, −0.75832861780128527288788226820, 0.75832861780128527288788226820, 2.11859904984605800539185865731, 2.19804625517662567576110363493, 3.07478254024297055431198054974, 3.28457437384958610690493771457, 4.11372971456978406634294044017, 4.51687053298683043767101964001, 5.26653230877033484086553748573, 5.33941418353216564672989958363, 5.54395070645542367514017902965, 6.26070345883133105806874898042, 6.73667846196512459043785934601, 7.16140182094123318922567456520, 7.54973587624073372361618498502, 7.961926639505284974667826140339, 8.523325819574670859769689729456, 9.086489278559170336108820841607, 9.630449055576491865947824382297, 9.995895005827938782847657328128, 10.07777529203296053239248113589

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