Properties

Label 4-1815e2-1.1-c1e2-0-10
Degree $4$
Conductor $3294225$
Sign $1$
Analytic cond. $210.042$
Root an. cond. $3.80694$
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  + 2·3-s − 4-s + 2·5-s + 9-s − 2·12-s + 4·15-s − 3·16-s − 2·20-s + 3·25-s − 4·27-s + 14·31-s − 36-s − 16·37-s + 2·45-s + 18·47-s − 6·48-s + 4·49-s + 6·53-s − 12·59-s − 4·60-s + 7·64-s + 2·67-s + 6·71-s + 6·75-s − 6·80-s − 11·81-s − 6·89-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s + 0.894·5-s + 1/3·9-s − 0.577·12-s + 1.03·15-s − 3/4·16-s − 0.447·20-s + 3/5·25-s − 0.769·27-s + 2.51·31-s − 1/6·36-s − 2.63·37-s + 0.298·45-s + 2.62·47-s − 0.866·48-s + 4/7·49-s + 0.824·53-s − 1.56·59-s − 0.516·60-s + 7/8·64-s + 0.244·67-s + 0.712·71-s + 0.692·75-s − 0.670·80-s − 1.22·81-s − 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3294225\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(210.042\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3294225} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3294225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.383274952\)
\(L(\frac12)\) \(\approx\) \(3.383274952\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( 1 - 2 T + p T^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 28 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
73$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−7.49098539101166052381138121357, −7.14703542001175774353519888820, −6.74818164566120742574663576809, −6.29509018904608682994380414516, −5.83306727345807364409216570044, −5.39118208869250254857641721193, −4.97205368792216796302593140136, −4.47309452929910871749352060787, −4.03150232891126431605696030941, −3.57372290050761653531974714460, −2.92612840452621368318519794069, −2.57507624467624512088497102936, −2.12891273114152411316998270488, −1.51135865206012438638246386158, −0.63628903793762127103374436332, 0.63628903793762127103374436332, 1.51135865206012438638246386158, 2.12891273114152411316998270488, 2.57507624467624512088497102936, 2.92612840452621368318519794069, 3.57372290050761653531974714460, 4.03150232891126431605696030941, 4.47309452929910871749352060787, 4.97205368792216796302593140136, 5.39118208869250254857641721193, 5.83306727345807364409216570044, 6.29509018904608682994380414516, 6.74818164566120742574663576809, 7.14703542001175774353519888820, 7.49098539101166052381138121357

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