Properties

Label 4-136e2-1.1-c0e2-0-0
Degree $4$
Conductor $18496$
Sign $1$
Analytic cond. $0.00460672$
Root an. cond. $0.260524$
Motivic weight $0$
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  − 2·3-s − 4-s + 2·9-s + 2·11-s + 2·12-s + 16-s − 2·27-s − 4·33-s − 2·36-s − 2·41-s − 2·44-s − 2·48-s − 64-s − 2·73-s + 3·81-s + 2·97-s + 4·99-s + 2·107-s + 2·108-s − 2·113-s + 2·121-s + 4·123-s + 127-s + 131-s + 4·132-s + 137-s + 139-s + ⋯
L(s)  = 1  − 2·3-s − 4-s + 2·9-s + 2·11-s + 2·12-s + 16-s − 2·27-s − 4·33-s − 2·36-s − 2·41-s − 2·44-s − 2·48-s − 64-s − 2·73-s + 3·81-s + 2·97-s + 4·99-s + 2·107-s + 2·108-s − 2·113-s + 2·121-s + 4·123-s + 127-s + 131-s + 4·132-s + 137-s + 139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(18496\)    =    \(2^{6} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(0.00460672\)
Root analytic conductor: \(0.260524\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{136} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 18496,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2024612097\)
\(L(\frac12)\) \(\approx\) \(0.2024612097\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
17$C_2$ \( 1 + T^{2} \)
good3$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
5$C_2^2$ \( 1 + T^{4} \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_2^2$ \( 1 + T^{4} \)
31$C_2^2$ \( 1 + T^{4} \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
79$C_2^2$ \( 1 + T^{4} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
97$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
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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

−14.10829379018122111854593772233, −12.85715835520574200022236771097, −12.83260234817688541120987881577, −11.77449855381196807410731862227, −11.76561960022422668253953730470, −11.58875341989955402464879387225, −10.57382924601604411035752812235, −10.31820460079360129722653774897, −9.663511030670705522782666378170, −9.083110844705150311940738088582, −8.718488301401035354803204082389, −7.80649168776885473376423669888, −7.09243364214847237611653603262, −6.35973643123559432573863254683, −6.15309186342952232387332769817, −5.33625654831850659557344740679, −4.89202963436691088873700380710, −4.12555615364533978441904967912, −3.55644053736731572668032407830, −1.42085106340463447177544392325, 1.42085106340463447177544392325, 3.55644053736731572668032407830, 4.12555615364533978441904967912, 4.89202963436691088873700380710, 5.33625654831850659557344740679, 6.15309186342952232387332769817, 6.35973643123559432573863254683, 7.09243364214847237611653603262, 7.80649168776885473376423669888, 8.718488301401035354803204082389, 9.083110844705150311940738088582, 9.663511030670705522782666378170, 10.31820460079360129722653774897, 10.57382924601604411035752812235, 11.58875341989955402464879387225, 11.76561960022422668253953730470, 11.77449855381196807410731862227, 12.83260234817688541120987881577, 12.85715835520574200022236771097, 14.10829379018122111854593772233

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