Properties

Label 4-2352e2-1.1-c1e2-0-13
Degree $4$
Conductor $5531904$
Sign $1$
Analytic cond. $352.718$
Root an. cond. $4.33368$
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  − 3-s − 3·5-s − 3·11-s + 3·15-s − 6·17-s + 2·19-s + 25-s + 27-s + 18·29-s − 5·31-s + 3·33-s − 10·37-s − 12·47-s + 6·51-s + 9·53-s + 9·55-s − 2·57-s + 9·59-s + 24·67-s + 12·73-s − 75-s − 9·79-s − 81-s + 6·83-s + 18·85-s − 18·87-s + 6·89-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s − 0.904·11-s + 0.774·15-s − 1.45·17-s + 0.458·19-s + 1/5·25-s + 0.192·27-s + 3.34·29-s − 0.898·31-s + 0.522·33-s − 1.64·37-s − 1.75·47-s + 0.840·51-s + 1.23·53-s + 1.21·55-s − 0.264·57-s + 1.17·59-s + 2.93·67-s + 1.40·73-s − 0.115·75-s − 1.01·79-s − 1/9·81-s + 0.658·83-s + 1.95·85-s − 1.92·87-s + 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5531904\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(352.718\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5531904,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8523812439\)
\(L(\frac12)\) \(\approx\) \(0.8523812439\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 + T + T^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 24 T + 259 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 12 T + 121 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 6 T + 101 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
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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

−8.870358082145670571259780866552, −8.756327652377679620402074960724, −8.319630403852854032092117916114, −8.155156944134884325638296327909, −7.66034326653501779110874636638, −7.19806901918613571728673618656, −6.78081907778318147192799522950, −6.67797606702674631144498205572, −6.11619950060724635160308281746, −5.59088040331932855406511781194, −5.01518628788412728756419260601, −4.89949716431670803921462181651, −4.48185062284219992318060024646, −3.94338253428313960853257266351, −3.33540839094935822959404921408, −3.24472425020464416750413971669, −2.20278394035975240260261568231, −2.17958298347073021326386965882, −0.907964854197658979566790830164, −0.42031193331433056464883398447, 0.42031193331433056464883398447, 0.907964854197658979566790830164, 2.17958298347073021326386965882, 2.20278394035975240260261568231, 3.24472425020464416750413971669, 3.33540839094935822959404921408, 3.94338253428313960853257266351, 4.48185062284219992318060024646, 4.89949716431670803921462181651, 5.01518628788412728756419260601, 5.59088040331932855406511781194, 6.11619950060724635160308281746, 6.67797606702674631144498205572, 6.78081907778318147192799522950, 7.19806901918613571728673618656, 7.66034326653501779110874636638, 8.155156944134884325638296327909, 8.319630403852854032092117916114, 8.756327652377679620402074960724, 8.870358082145670571259780866552

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