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Properties

Label 4-2312e2-1.1-c0e2-0-2
Degree $4$
Conductor $5345344$
Sign $1$
Analytic cond. $1.33134$
Root an. cond. $1.07416$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 5345344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5345344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5345344\)    =    \(2^{6} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(1.33134\)
Root analytic conductor: \(1.07416\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5345344,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.747664696\)
\(L(\frac12)\) \(\approx\) \(1.747664696\)
\(L(1)\) 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 \( 1 \)
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

−9.578716163200738834477334255141, −8.879878327375682117319806791480, −8.660174787931424491343480036435, −8.075760031026605745351206774004, −8.061037241949871218661066524175, −7.76950220363082772312762822276, −7.33456217188006719694955678008, −6.86612811204264592196064233154, −6.28203709917504211777574357983, −5.74100164481167738636768386791, −5.28861005223960289897663629154, −5.01251587117738563151859986688, −4.43584965500702648356072977155, −4.10674858456214283759283665622, −3.56474124342709356998305094179, −3.14631737396415544365951699766, −2.62730330798888613834246385901, −2.49843459899612704812075948417, −1.77402779539533271587330371493, −0.821569623257130770884288497517, 0.821569623257130770884288497517, 1.77402779539533271587330371493, 2.49843459899612704812075948417, 2.62730330798888613834246385901, 3.14631737396415544365951699766, 3.56474124342709356998305094179, 4.10674858456214283759283665622, 4.43584965500702648356072977155, 5.01251587117738563151859986688, 5.28861005223960289897663629154, 5.74100164481167738636768386791, 6.28203709917504211777574357983, 6.86612811204264592196064233154, 7.33456217188006719694955678008, 7.76950220363082772312762822276, 8.061037241949871218661066524175, 8.075760031026605745351206774004, 8.660174787931424491343480036435, 8.879878327375682117319806791480, 9.578716163200738834477334255141

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