Properties

Label 4-84e4-1.1-c1e2-0-5
Degree $4$
Conductor $49787136$
Sign $1$
Analytic cond. $3174.47$
Root an. cond. $7.50616$
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  + 4·19-s + 7·25-s − 18·29-s − 10·31-s + 20·37-s + 24·47-s + 18·53-s − 18·59-s + 6·83-s + 8·103-s − 8·109-s − 12·113-s + 19·121-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 + ⋯
L(s)  = 1  + 0.917·19-s + 7/5·25-s − 3.34·29-s − 1.79·31-s + 3.28·37-s + 3.50·47-s + 2.47·53-s − 2.34·59-s + 0.658·83-s + 0.788·103-s − 0.766·109-s − 1.12·113-s + 1.72·121-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(49787136\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(3174.47\)
Root analytic conductor: \(7.50616\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7056} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 49787136,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.557143731\)
\(L(\frac12)\) \(\approx\) \(2.557143731\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 166 T^{2} + p^{2} T^{4} \)
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

−7.922805769723184275094165454926, −7.72347155552803628293197550886, −7.30242079572461138145011395745, −7.22674530021183876229445174046, −6.97862422849910674120208893947, −6.13237782380758189583492995173, −5.99221646179381375155257645569, −5.68338728388227704195333280308, −5.42451029678438750515486412429, −4.96626039315885766655515788357, −4.51421604574644646684376970045, −4.07290778153711235634192678815, −3.77193084854915222473909803013, −3.50563295897533306604851463727, −2.87349734061416486565478200306, −2.38946038519245983373218456130, −2.25358042614799358280280331150, −1.45501337233404181680203322840, −1.03810558623116636556249377159, −0.43812721032310367002082372579, 0.43812721032310367002082372579, 1.03810558623116636556249377159, 1.45501337233404181680203322840, 2.25358042614799358280280331150, 2.38946038519245983373218456130, 2.87349734061416486565478200306, 3.50563295897533306604851463727, 3.77193084854915222473909803013, 4.07290778153711235634192678815, 4.51421604574644646684376970045, 4.96626039315885766655515788357, 5.42451029678438750515486412429, 5.68338728388227704195333280308, 5.99221646179381375155257645569, 6.13237782380758189583492995173, 6.97862422849910674120208893947, 7.22674530021183876229445174046, 7.30242079572461138145011395745, 7.72347155552803628293197550886, 7.922805769723184275094165454926

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