Properties

Label 4-2240e2-1.1-c1e2-0-13
Degree $4$
Conductor $5017600$
Sign $1$
Analytic cond. $319.926$
Root an. cond. $4.22924$
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  + 4·5-s − 3·9-s + 6·11-s + 16·19-s + 11·25-s − 2·29-s + 4·31-s − 12·41-s − 12·45-s − 49-s + 24·55-s + 20·59-s − 14·79-s − 16·89-s + 64·95-s − 18·99-s + 24·101-s − 14·109-s + 5·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s + ⋯
L(s)  = 1  + 1.78·5-s − 9-s + 1.80·11-s + 3.67·19-s + 11/5·25-s − 0.371·29-s + 0.718·31-s − 1.87·41-s − 1.78·45-s − 1/7·49-s + 3.23·55-s + 2.60·59-s − 1.57·79-s − 1.69·89-s + 6.56·95-s − 1.80·99-s + 2.38·101-s − 1.34·109-s + 5/11·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5017600\)    =    \(2^{12} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(319.926\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2240} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5017600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.903263691\)
\(L(\frac12)\) \(\approx\) \(4.903263691\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 27 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 185 T^{2} + p^{2} T^{4} \)
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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

−9.444937754881889020976340612451, −8.806347243927272990309576059979, −8.743799939616966892473971278641, −8.174808188300188667863368179361, −7.63401815350136425103219077516, −7.16800269198948105149103308177, −6.69488843095879628863724483095, −6.68939003075776498415699966681, −5.83813556402404487514444157361, −5.82681068969927050468553979772, −5.27233249759591380841527085999, −5.14621923235676818267446416094, −4.53032506320539930054225881121, −3.64433858944728584870714096975, −3.55109348064170407627398046915, −2.81820823696117805048741695195, −2.63251548610733233969776936747, −1.68738533447412663463829885713, −1.36931380178288279734317856982, −0.850475093751458666498531711327, 0.850475093751458666498531711327, 1.36931380178288279734317856982, 1.68738533447412663463829885713, 2.63251548610733233969776936747, 2.81820823696117805048741695195, 3.55109348064170407627398046915, 3.64433858944728584870714096975, 4.53032506320539930054225881121, 5.14621923235676818267446416094, 5.27233249759591380841527085999, 5.82681068969927050468553979772, 5.83813556402404487514444157361, 6.68939003075776498415699966681, 6.69488843095879628863724483095, 7.16800269198948105149103308177, 7.63401815350136425103219077516, 8.174808188300188667863368179361, 8.743799939616966892473971278641, 8.806347243927272990309576059979, 9.444937754881889020976340612451

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