Properties

Label 4-1170e2-1.1-c1e2-0-15
Degree $4$
Conductor $1368900$
Sign $1$
Analytic cond. $87.2822$
Root an. cond. $3.05654$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 4·5-s − 4·11-s + 16-s + 8·19-s + 4·20-s + 11·25-s + 8·29-s + 16·31-s + 12·41-s + 4·44-s + 10·49-s + 16·55-s + 20·59-s − 28·61-s − 64-s + 8·71-s − 8·76-s + 16·79-s − 4·80-s + 12·89-s − 32·95-s − 11·100-s − 8·101-s + 36·109-s − 8·116-s − 10·121-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.78·5-s − 1.20·11-s + 1/4·16-s + 1.83·19-s + 0.894·20-s + 11/5·25-s + 1.48·29-s + 2.87·31-s + 1.87·41-s + 0.603·44-s + 10/7·49-s + 2.15·55-s + 2.60·59-s − 3.58·61-s − 1/8·64-s + 0.949·71-s − 0.917·76-s + 1.80·79-s − 0.447·80-s + 1.27·89-s − 3.28·95-s − 1.09·100-s − 0.796·101-s + 3.44·109-s − 0.742·116-s − 0.909·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1368900\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(87.2822\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1170} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1368900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.387948091\)
\(L(\frac12)\) \(\approx\) \(1.387948091\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
13$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
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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

−9.979970515295386925813278834114, −9.654174481263098400305920372138, −8.997031072783284288748607379997, −8.744488276021039587369213120821, −8.171815759576235563455702877760, −7.977949416082132357355638043698, −7.53615272390464044425244229702, −7.42680077058700292471204770823, −6.70138623465424849284267852003, −6.28144202217683831298153005352, −5.69199562201546320598422739492, −5.12909958999602973102193775545, −4.63486277788420801135174057666, −4.56485444308437704603388022604, −3.84389128685193229933124713034, −3.34827504757729037250009898440, −2.81451474276305788264398277562, −2.50507709629848732940770638817, −0.930918142397487302835950278504, −0.72864149967538039579016075059, 0.72864149967538039579016075059, 0.930918142397487302835950278504, 2.50507709629848732940770638817, 2.81451474276305788264398277562, 3.34827504757729037250009898440, 3.84389128685193229933124713034, 4.56485444308437704603388022604, 4.63486277788420801135174057666, 5.12909958999602973102193775545, 5.69199562201546320598422739492, 6.28144202217683831298153005352, 6.70138623465424849284267852003, 7.42680077058700292471204770823, 7.53615272390464044425244229702, 7.977949416082132357355638043698, 8.171815759576235563455702877760, 8.744488276021039587369213120821, 8.997031072783284288748607379997, 9.654174481263098400305920372138, 9.979970515295386925813278834114

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