Properties

Label 4-4112-1.1-c1e2-0-0
Degree $4$
Conductor $4112$
Sign $1$
Analytic cond. $0.262184$
Root an. cond. $0.715569$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 3·8-s + 2·9-s + 4·13-s − 16-s + 4·17-s − 2·18-s − 6·25-s − 4·26-s − 12·29-s − 5·32-s − 4·34-s − 2·36-s − 16·37-s + 12·41-s + 2·49-s + 6·50-s − 4·52-s + 8·53-s + 12·58-s + 4·61-s + 7·64-s − 4·68-s + 6·72-s − 12·73-s + 16·74-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.06·8-s + 2/3·9-s + 1.10·13-s − 1/4·16-s + 0.970·17-s − 0.471·18-s − 6/5·25-s − 0.784·26-s − 2.22·29-s − 0.883·32-s − 0.685·34-s − 1/3·36-s − 2.63·37-s + 1.87·41-s + 2/7·49-s + 0.848·50-s − 0.554·52-s + 1.09·53-s + 1.57·58-s + 0.512·61-s + 7/8·64-s − 0.485·68-s + 0.707·72-s − 1.40·73-s + 1.85·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4112 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4112 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4112\)    =    \(2^{4} \cdot 257\)
Sign: $1$
Analytic conductor: \(0.262184\)
Root analytic conductor: \(0.715569\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4112,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5440100461\)
\(L(\frac12)\) \(\approx\) \(0.5440100461\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + p T^{2} \)
257$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−12.55113911913291432504173264338, −11.85008133515623933385800153806, −11.05139471712736036943594714001, −10.67459149203692243797489835816, −9.861251387498423866382206694842, −9.610791028647234859608006801501, −8.765669568456885891616139520575, −8.371288656108001150737374927218, −7.39835362300314054472468513894, −7.24477666504451448204354780334, −5.87545876078630608591993295901, −5.41426981374372084835653925495, −4.16331826282344293868388099851, −3.63582255783349937241895391884, −1.64408879948474926792686178522, 1.64408879948474926792686178522, 3.63582255783349937241895391884, 4.16331826282344293868388099851, 5.41426981374372084835653925495, 5.87545876078630608591993295901, 7.24477666504451448204354780334, 7.39835362300314054472468513894, 8.371288656108001150737374927218, 8.765669568456885891616139520575, 9.610791028647234859608006801501, 9.861251387498423866382206694842, 10.67459149203692243797489835816, 11.05139471712736036943594714001, 11.85008133515623933385800153806, 12.55113911913291432504173264338

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