Properties

Label 4-6272-1.1-c1e2-0-3
Degree $4$
Conductor $6272$
Sign $1$
Analytic cond. $0.399908$
Root an. cond. $0.795225$
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  + 2·9-s + 2·25-s − 4·29-s − 20·37-s − 7·49-s + 12·53-s − 5·81-s − 4·109-s + 20·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 18·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 2/3·9-s + 2/5·25-s − 0.742·29-s − 3.28·37-s − 49-s + 1.64·53-s − 5/9·81-s − 0.383·109-s + 1.88·113-s − 0.545·121-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 + 1.38·169-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 6272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6272\)    =    \(2^{7} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.399908\)
Root analytic conductor: \(0.795225\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{6272} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6272,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9073050183\)
\(L(\frac12)\) \(\approx\) \(0.9073050183\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
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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

−12.08890731676568690871747441950, −11.44186221894017102496655373262, −10.80763763733693351133636026312, −10.24149401345816652698031438464, −9.836462663224188369892012114703, −8.944319633591112961500388380260, −8.626710534180408615665199670513, −7.74063922969685639448832851767, −7.08621645387800538550255481789, −6.64901273400801050315809490253, −5.62561300264421634210733970294, −5.02174775184510003203595696449, −4.08201528878263870261914412581, −3.25431794901031101617142408905, −1.83712417196464390825545201827, 1.83712417196464390825545201827, 3.25431794901031101617142408905, 4.08201528878263870261914412581, 5.02174775184510003203595696449, 5.62561300264421634210733970294, 6.64901273400801050315809490253, 7.08621645387800538550255481789, 7.74063922969685639448832851767, 8.626710534180408615665199670513, 8.944319633591112961500388380260, 9.836462663224188369892012114703, 10.24149401345816652698031438464, 10.80763763733693351133636026312, 11.44186221894017102496655373262, 12.08890731676568690871747441950

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