Properties

Label 4-112e2-1.1-c2e2-0-0
Degree $4$
Conductor $12544$
Sign $1$
Analytic cond. $9.31335$
Root an. cond. $1.74693$
Motivic weight $2$
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  − 10·7-s − 6·9-s + 12·11-s + 60·23-s + 26·25-s − 12·29-s + 20·37-s − 20·43-s + 51·49-s + 180·53-s + 60·63-s + 140·67-s − 84·71-s − 120·77-s − 148·79-s − 45·81-s − 72·99-s + 300·107-s − 172·109-s + 180·113-s − 134·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1.42·7-s − 2/3·9-s + 1.09·11-s + 2.60·23-s + 1.03·25-s − 0.413·29-s + 0.540·37-s − 0.465·43-s + 1.04·49-s + 3.39·53-s + 0.952·63-s + 2.08·67-s − 1.18·71-s − 1.55·77-s − 1.87·79-s − 5/9·81-s − 0.727·99-s + 2.80·107-s − 1.57·109-s + 1.59·113-s − 1.10·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12544\)    =    \(2^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(9.31335\)
Root analytic conductor: \(1.74693\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{112} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12544,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.386270835\)
\(L(\frac12)\) \(\approx\) \(1.386270835\)
\(L(2)\) 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 \( 1 \)
7$C_2$ \( 1 + 10 T + p^{2} T^{2} \)
good3$C_2^2$ \( 1 + 2 p T^{2} + p^{4} T^{4} \)
5$C_2^2$ \( 1 - 26 T^{2} + p^{4} T^{4} \)
11$C_2$ \( ( 1 - 6 T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 314 T^{2} + p^{4} T^{4} \)
17$C_2^2$ \( 1 - 194 T^{2} + p^{4} T^{4} \)
19$C_2^2$ \( 1 - 122 T^{2} + p^{4} T^{4} \)
23$C_2$ \( ( 1 - 30 T + p^{2} T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p^{2} T^{2} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
37$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 962 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 10 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 4034 T^{2} + p^{4} T^{4} \)
53$C_2$ \( ( 1 - 90 T + p^{2} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 6362 T^{2} + p^{4} T^{4} \)
61$C_2^2$ \( 1 - 6842 T^{2} + p^{4} T^{4} \)
67$C_2$ \( ( 1 - 70 T + p^{2} T^{2} )^{2} \)
71$C_2$ \( ( 1 + 42 T + p^{2} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 958 T^{2} + p^{4} T^{4} \)
79$C_2$ \( ( 1 + 74 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 9722 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 + 5758 T^{2} + p^{4} T^{4} \)
97$C_2^2$ \( 1 - 12674 T^{2} + p^{4} T^{4} \)
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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

−14.17808117320178025643885352170, −13.06606445981340314740589351753, −12.76002063979252544924595205489, −12.13333627057238922726449176364, −11.45029810542154833913640141068, −11.21790886123752500207346865186, −10.42316649877073761907216020259, −9.920152223235230301218490548721, −9.301948613517601958553394696236, −8.723709063513081752686972045586, −8.665147378184032865125193912401, −7.27443939676604785855314913089, −7.01392112553932178190743770539, −6.43753416918624150168726580636, −5.73459852618805461372178183945, −5.05591090448400625333918188151, −4.05583096627208211987594464420, −3.28095790633438729225356922982, −2.66246405893939247565094900351, −0.907563174937711343018619709148, 0.907563174937711343018619709148, 2.66246405893939247565094900351, 3.28095790633438729225356922982, 4.05583096627208211987594464420, 5.05591090448400625333918188151, 5.73459852618805461372178183945, 6.43753416918624150168726580636, 7.01392112553932178190743770539, 7.27443939676604785855314913089, 8.665147378184032865125193912401, 8.723709063513081752686972045586, 9.301948613517601958553394696236, 9.920152223235230301218490548721, 10.42316649877073761907216020259, 11.21790886123752500207346865186, 11.45029810542154833913640141068, 12.13333627057238922726449176364, 12.76002063979252544924595205489, 13.06606445981340314740589351753, 14.17808117320178025643885352170

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