Properties

Label 4-2112e2-1.1-c3e2-0-1
Degree $4$
Conductor $4460544$
Sign $1$
Analytic cond. $15528.1$
Root an. cond. $11.1629$
Motivic weight $3$
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  − 6·3-s − 16·5-s + 2·7-s + 27·9-s − 22·11-s + 76·13-s + 96·15-s − 26·17-s + 54·19-s − 12·21-s + 224·23-s + 74·25-s − 108·27-s − 222·29-s − 40·31-s + 132·33-s − 32·35-s + 48·37-s − 456·39-s − 494·41-s + 66·43-s − 432·45-s − 64·47-s − 650·49-s + 156·51-s + 84·53-s + 352·55-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.43·5-s + 0.107·7-s + 9-s − 0.603·11-s + 1.62·13-s + 1.65·15-s − 0.370·17-s + 0.652·19-s − 0.124·21-s + 2.03·23-s + 0.591·25-s − 0.769·27-s − 1.42·29-s − 0.231·31-s + 0.696·33-s − 0.154·35-s + 0.213·37-s − 1.87·39-s − 1.88·41-s + 0.234·43-s − 1.43·45-s − 0.198·47-s − 1.89·49-s + 0.428·51-s + 0.217·53-s + 0.862·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4460544\)    =    \(2^{12} \cdot 3^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(15528.1\)
Root analytic conductor: \(11.1629\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4460544,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.219664603\)
\(L(\frac12)\) \(\approx\) \(1.219664603\)
\(L(\frac{5}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 + p T )^{2} \)
11$C_1$ \( ( 1 + p T )^{2} \)
good5$D_{4}$ \( 1 + 16 T + 182 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 - 2 T + 654 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 76 T + 5310 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 26 T + 2570 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 54 T + 11774 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2$ \( ( 1 - 112 T + p^{3} T^{2} )^{2} \)
29$D_{4}$ \( 1 + 222 T + 43642 T^{2} + 222 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 40 T - 29250 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 48 T + 85910 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 494 T + 198818 T^{2} + 494 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 66 T + 99086 T^{2} - 66 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 64 T + 189662 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 84 T + 164350 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2$ \( ( 1 + 196 T + p^{3} T^{2} )^{2} \)
61$D_{4}$ \( 1 - 1104 T + 736358 T^{2} - 1104 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 928 T + 626214 T^{2} + 928 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 456 T + 488494 T^{2} - 456 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 592 T + 341742 T^{2} + 592 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 230 T + 954126 T^{2} + 230 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 348 T + 307798 T^{2} + 348 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 972 T + 1645606 T^{2} - 972 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 1184 T + 720510 T^{2} + 1184 p^{3} T^{3} + p^{6} T^{4} \)
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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

−8.815810639442874799172705007753, −8.518844064734792291814273750273, −8.079170676970730843229387032713, −7.82227227478686751033263048979, −7.20899282042960774437768775724, −7.09770802729487309605554086625, −6.61990733007658328935094938459, −6.26021005530900535605219196833, −5.57833888984777959040823979141, −5.45385926001313293068438342322, −4.94777553986942266350621156727, −4.58361105548085528070237480827, −3.96717075587333736444893185392, −3.77420062097868305644775264830, −3.11460794116014036098633868459, −2.94376281174833214094834403519, −1.65220505299215808183299885422, −1.59592044525042769911340118127, −0.65723672139089052886033253474, −0.39155561342061303224272842288, 0.39155561342061303224272842288, 0.65723672139089052886033253474, 1.59592044525042769911340118127, 1.65220505299215808183299885422, 2.94376281174833214094834403519, 3.11460794116014036098633868459, 3.77420062097868305644775264830, 3.96717075587333736444893185392, 4.58361105548085528070237480827, 4.94777553986942266350621156727, 5.45385926001313293068438342322, 5.57833888984777959040823979141, 6.26021005530900535605219196833, 6.61990733007658328935094938459, 7.09770802729487309605554086625, 7.20899282042960774437768775724, 7.82227227478686751033263048979, 8.079170676970730843229387032713, 8.518844064734792291814273750273, 8.815810639442874799172705007753

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