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Properties

Label 4-104e2-1.1-c1e2-0-9
Degree $4$
Conductor $10816$
Sign $1$
Analytic cond. $0.689637$
Root an. cond. $0.911287$
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 + 2·4-s − 2·5-s + 5·9-s − 4·10-s − 4·11-s + 6·13-s − 4·16-s + 6·17-s + 10·18-s − 4·20-s − 8·22-s − 12·23-s − 7·25-s + 12·26-s − 8·32-s + 12·34-s + 10·36-s − 6·37-s − 8·44-s − 10·45-s − 24·46-s + 5·49-s − 14·50-s + 12·52-s + 8·55-s − 20·59-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 0.894·5-s + 5/3·9-s − 1.26·10-s − 1.20·11-s + 1.66·13-s − 16-s + 1.45·17-s + 2.35·18-s − 0.894·20-s − 1.70·22-s − 2.50·23-s − 7/5·25-s + 2.35·26-s − 1.41·32-s + 2.05·34-s + 5/3·36-s − 0.986·37-s − 1.20·44-s − 1.49·45-s − 3.53·46-s + 5/7·49-s − 1.97·50-s + 1.66·52-s + 1.07·55-s − 2.60·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10816 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10816\)    =    \(2^{6} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.689637\)
Root analytic conductor: \(0.911287\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10816,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.804631911\)
\(L(\frac12)\) \(\approx\) \(1.804631911\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
13$C_2$ \( 1 - 6 T + p T^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 117 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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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

−13.95530504425463177219684818725, −13.37173768154440220279298237117, −13.27491948009287716451620243992, −12.36455323142233149799899267619, −12.13823934744062902284794236554, −11.87880212660656222739612614403, −10.84737676313181404484593279239, −10.63166485054981770210982109794, −9.858004989560526093512736265896, −9.416061035242936798259608912489, −8.110565670284822805233160735207, −8.042626235264431202428795382995, −7.41462406189326598592223265204, −6.51233162850101765397403798126, −5.92452216695234809845577081561, −5.33318281045966834656803326883, −4.43460414278886521159715690280, −3.72887635245383174516693223809, −3.59233905038738281641359546513, −1.97087923560232949779890484285, 1.97087923560232949779890484285, 3.59233905038738281641359546513, 3.72887635245383174516693223809, 4.43460414278886521159715690280, 5.33318281045966834656803326883, 5.92452216695234809845577081561, 6.51233162850101765397403798126, 7.41462406189326598592223265204, 8.042626235264431202428795382995, 8.110565670284822805233160735207, 9.416061035242936798259608912489, 9.858004989560526093512736265896, 10.63166485054981770210982109794, 10.84737676313181404484593279239, 11.87880212660656222739612614403, 12.13823934744062902284794236554, 12.36455323142233149799899267619, 13.27491948009287716451620243992, 13.37173768154440220279298237117, 13.95530504425463177219684818725

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