Properties

Degree $4$
Conductor $64545156$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 3·4-s − 4·5-s + 4·6-s + 3·7-s − 4·8-s + 3·9-s + 8·10-s − 11-s − 6·12-s − 2·13-s − 6·14-s + 8·15-s + 5·16-s + 11·17-s − 6·18-s − 4·19-s − 12·20-s − 6·21-s + 2·22-s − 2·23-s + 8·24-s + 2·25-s + 4·26-s − 4·27-s + 9·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.78·5-s + 1.63·6-s + 1.13·7-s − 1.41·8-s + 9-s + 2.52·10-s − 0.301·11-s − 1.73·12-s − 0.554·13-s − 1.60·14-s + 2.06·15-s + 5/4·16-s + 2.66·17-s − 1.41·18-s − 0.917·19-s − 2.68·20-s − 1.30·21-s + 0.426·22-s − 0.417·23-s + 1.63·24-s + 2/5·25-s + 0.784·26-s − 0.769·27-s + 1.70·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64545156 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64545156 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(64545156\)    =    \(2^{2} \cdot 3^{2} \cdot 13^{2} \cdot 103^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{8034} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 64545156,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
103$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$D_{4}$ \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 + 6 T + 54 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 10 T + 82 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 9 T + 122 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 11 T + 126 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 5 T + 114 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 10 T + 30 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−7.61504821740299154111283982069, −7.44612128071544665986765105257, −7.20480031889648990049588348652, −6.85137825167425043529220644590, −6.33920176261175380405005760155, −6.01634244011360865153011104285, −5.54789648427194112280809942946, −5.31751596245021399152497590244, −4.85432041254355892840383758923, −4.60151109825343086580730462966, −4.02940192310934551562604174216, −3.78794597914259642695522924027, −3.10912921376212607483798009680, −3.10018059791519605389740493651, −2.10981902749199411924745735354, −1.84068298358037682276259880991, −1.18486051978851374158908330869, −0.909462209393519328707906409985, 0, 0, 0.909462209393519328707906409985, 1.18486051978851374158908330869, 1.84068298358037682276259880991, 2.10981902749199411924745735354, 3.10018059791519605389740493651, 3.10912921376212607483798009680, 3.78794597914259642695522924027, 4.02940192310934551562604174216, 4.60151109825343086580730462966, 4.85432041254355892840383758923, 5.31751596245021399152497590244, 5.54789648427194112280809942946, 6.01634244011360865153011104285, 6.33920176261175380405005760155, 6.85137825167425043529220644590, 7.20480031889648990049588348652, 7.44612128071544665986765105257, 7.61504821740299154111283982069

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