Properties

Label 4-306e2-1.1-c1e2-0-3
Degree $4$
Conductor $93636$
Sign $1$
Analytic cond. $5.97031$
Root an. cond. $1.56314$
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 + 3·4-s − 4·8-s + 4·13-s + 5·16-s + 6·17-s + 4·19-s + 8·25-s − 8·26-s − 6·32-s − 12·34-s − 8·38-s − 8·43-s + 24·47-s − 4·49-s − 16·50-s + 12·52-s − 12·53-s − 12·59-s + 7·64-s − 20·67-s + 18·68-s + 12·76-s + 12·83-s + 16·86-s − 12·89-s − 48·94-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s + 1.10·13-s + 5/4·16-s + 1.45·17-s + 0.917·19-s + 8/5·25-s − 1.56·26-s − 1.06·32-s − 2.05·34-s − 1.29·38-s − 1.21·43-s + 3.50·47-s − 4/7·49-s − 2.26·50-s + 1.66·52-s − 1.64·53-s − 1.56·59-s + 7/8·64-s − 2.44·67-s + 2.18·68-s + 1.37·76-s + 1.31·83-s + 1.72·86-s − 1.27·89-s − 4.95·94-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93636 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93636 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(93636\)    =    \(2^{2} \cdot 3^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(5.97031\)
Root analytic conductor: \(1.56314\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{306} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 93636,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9071732046\)
\(L(\frac12)\) \(\approx\) \(0.9071732046\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
17$C_2$ \( 1 - 6 T + p T^{2} \)
good5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 56 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 140 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 122 T^{2} + p^{2} T^{4} \)
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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

−12.02620659240219112526598306365, −11.13230241228995560910523046248, −10.88063574061265581732493567012, −10.63551944026625307654854089393, −9.990874840833729381787997709838, −9.527277597510449659511349470592, −9.178900121570348977684715890071, −8.651352448882167569363164085531, −8.198354951513115824164346223962, −7.77879977604317506683805015754, −7.17694207720659108003448634618, −6.88595554935714671088420439183, −5.98832854861153248731297336061, −5.81211010404588205818654087297, −5.02455124867949836283567224813, −4.17281046156135786819454367902, −3.11140308465618835864316615034, −3.04551675307913225849244835951, −1.63882728095410674323939409728, −0.987150374847621157800648176480, 0.987150374847621157800648176480, 1.63882728095410674323939409728, 3.04551675307913225849244835951, 3.11140308465618835864316615034, 4.17281046156135786819454367902, 5.02455124867949836283567224813, 5.81211010404588205818654087297, 5.98832854861153248731297336061, 6.88595554935714671088420439183, 7.17694207720659108003448634618, 7.77879977604317506683805015754, 8.198354951513115824164346223962, 8.651352448882167569363164085531, 9.178900121570348977684715890071, 9.527277597510449659511349470592, 9.990874840833729381787997709838, 10.63551944026625307654854089393, 10.88063574061265581732493567012, 11.13230241228995560910523046248, 12.02620659240219112526598306365

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