Properties

Label 4-5445e2-1.1-c1e2-0-7
Degree $4$
Conductor $29648025$
Sign $1$
Analytic cond. $1890.38$
Root an. cond. $6.59382$
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 + 4-s + 2·5-s + 4·7-s + 4·10-s + 8·13-s + 8·14-s + 16-s + 8·17-s + 2·20-s + 3·25-s + 16·26-s + 4·28-s + 4·29-s − 2·32-s + 16·34-s + 8·35-s − 4·37-s + 12·41-s + 12·43-s − 2·49-s + 6·50-s + 8·52-s − 12·53-s + 8·58-s + 8·59-s − 4·61-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s + 0.894·5-s + 1.51·7-s + 1.26·10-s + 2.21·13-s + 2.13·14-s + 1/4·16-s + 1.94·17-s + 0.447·20-s + 3/5·25-s + 3.13·26-s + 0.755·28-s + 0.742·29-s − 0.353·32-s + 2.74·34-s + 1.35·35-s − 0.657·37-s + 1.87·41-s + 1.82·43-s − 2/7·49-s + 0.848·50-s + 1.10·52-s − 1.64·53-s + 1.05·58-s + 1.04·59-s − 0.512·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29648025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29648025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(29648025\)    =    \(3^{4} \cdot 5^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(1890.38\)
Root analytic conductor: \(6.59382\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5445} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 29648025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(14.46301587\)
\(L(\frac12)\) \(\approx\) \(14.46301587\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
good2$D_{4}$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 154 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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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

−8.184634255285142865140303647380, −7.85954057968084156005808612198, −7.82935500034133719715214491057, −7.30097239134651260879117695149, −6.65921136390532920334326821949, −6.43440016195920616082350539486, −5.92011876838888545158631286331, −5.75213654005785566336985750282, −5.33720244303735127442092987412, −5.23856763699651031759734012370, −4.64144588106317236533705950964, −4.38992513739548570022790784955, −3.91424476464776678903641097137, −3.69795996150534809220635861927, −3.12889975732287872040313663241, −2.81908782308141833850290925488, −2.13364715184755587713038158870, −1.54984122971752724080285589921, −1.26734886930746115142007972380, −0.894211443976235578482143057558, 0.894211443976235578482143057558, 1.26734886930746115142007972380, 1.54984122971752724080285589921, 2.13364715184755587713038158870, 2.81908782308141833850290925488, 3.12889975732287872040313663241, 3.69795996150534809220635861927, 3.91424476464776678903641097137, 4.38992513739548570022790784955, 4.64144588106317236533705950964, 5.23856763699651031759734012370, 5.33720244303735127442092987412, 5.75213654005785566336985750282, 5.92011876838888545158631286331, 6.43440016195920616082350539486, 6.65921136390532920334326821949, 7.30097239134651260879117695149, 7.82935500034133719715214491057, 7.85954057968084156005808612198, 8.184634255285142865140303647380

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