Properties

Label 4-332928-1.1-c1e2-0-9
Degree $4$
Conductor $332928$
Sign $1$
Analytic cond. $21.2277$
Root an. cond. $2.14647$
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 + 3-s + 4-s − 6-s − 8-s − 2·9-s + 9·11-s + 12-s + 16-s + 2·18-s + 19-s − 9·22-s − 24-s − 7·25-s − 5·27-s − 32-s + 9·33-s − 2·36-s − 38-s − 3·41-s − 2·43-s + 9·44-s + 48-s − 4·49-s + 7·50-s + 5·54-s + 57-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s − 2/3·9-s + 2.71·11-s + 0.288·12-s + 1/4·16-s + 0.471·18-s + 0.229·19-s − 1.91·22-s − 0.204·24-s − 7/5·25-s − 0.962·27-s − 0.176·32-s + 1.56·33-s − 1/3·36-s − 0.162·38-s − 0.468·41-s − 0.304·43-s + 1.35·44-s + 0.144·48-s − 4/7·49-s + 0.989·50-s + 0.680·54-s + 0.132·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(332928\)    =    \(2^{7} \cdot 3^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(21.2277\)
Root analytic conductor: \(2.14647\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 332928,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.713073095\)
\(L(\frac12)\) \(\approx\) \(1.713073095\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
3$C_2$ \( 1 - T + p T^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 49 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 104 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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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

−8.822886882108312347430863500518, −8.297364303929656969893857581327, −8.101963104451712253135223609141, −7.39844126782035767676833035798, −6.94177298315197469872909697055, −6.46700268473231076500275389112, −6.13245652555746165211403471082, −5.58946294533866469920253156776, −4.86226144548405919991692719216, −4.10459018903400883921335269339, −3.56219503035160071188740827045, −3.36801176805834742666903497601, −2.22698293189520995154230422732, −1.81000315951001215376074196057, −0.843425683843677716281309704308, 0.843425683843677716281309704308, 1.81000315951001215376074196057, 2.22698293189520995154230422732, 3.36801176805834742666903497601, 3.56219503035160071188740827045, 4.10459018903400883921335269339, 4.86226144548405919991692719216, 5.58946294533866469920253156776, 6.13245652555746165211403471082, 6.46700268473231076500275389112, 6.94177298315197469872909697055, 7.39844126782035767676833035798, 8.101963104451712253135223609141, 8.297364303929656969893857581327, 8.822886882108312347430863500518

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