Properties

Label 4-2736e2-1.1-c1e2-0-11
Degree $4$
Conductor $7485696$
Sign $1$
Analytic cond. $477.294$
Root an. cond. $4.67408$
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  + 2·5-s − 14·17-s − 7·25-s − 5·49-s + 30·61-s + 22·73-s − 28·85-s + 20·101-s + 3·121-s − 26·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 0.894·5-s − 3.39·17-s − 7/5·25-s − 5/7·49-s + 3.84·61-s + 2.57·73-s − 3.03·85-s + 1.99·101-s + 3/11·121-s − 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7485696\)    =    \(2^{8} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(477.294\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2736} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7485696,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.342901016\)
\(L(\frac12)\) \(\approx\) \(1.342901016\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - p T^{2} )^{2} \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2$ \( ( 1 - p T^{2} )^{2} \)
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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

−9.253722077343463584061172115274, −8.789203814468410657458081175564, −8.242811409315443638822099337603, −8.020252288282131077617668391171, −7.48234465963839701103035147954, −6.83371142556904513206406479624, −6.82688651803480569959002336625, −6.32631396517094626619036615991, −6.08535659431749085987035216836, −5.57329956825449175714035398194, −4.95948616662942962246825509142, −4.91508712412840842965599357449, −4.24759561997224327573187415089, −3.78582158723682244956314339667, −3.59742293432366150954159283400, −2.50702635510614936132097004729, −2.25068134317812301064841386519, −2.15851289610000563168969067959, −1.33622192729485105473819296935, −0.36173015810441053873242161007, 0.36173015810441053873242161007, 1.33622192729485105473819296935, 2.15851289610000563168969067959, 2.25068134317812301064841386519, 2.50702635510614936132097004729, 3.59742293432366150954159283400, 3.78582158723682244956314339667, 4.24759561997224327573187415089, 4.91508712412840842965599357449, 4.95948616662942962246825509142, 5.57329956825449175714035398194, 6.08535659431749085987035216836, 6.32631396517094626619036615991, 6.82688651803480569959002336625, 6.83371142556904513206406479624, 7.48234465963839701103035147954, 8.020252288282131077617668391171, 8.242811409315443638822099337603, 8.789203814468410657458081175564, 9.253722077343463584061172115274

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