Properties

Label 4-98e2-1.1-c3e2-0-1
Degree $4$
Conductor $9604$
Sign $1$
Analytic cond. $33.4336$
Root an. cond. $2.40461$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 12·4-s − 32·8-s − 4·9-s − 28·11-s + 80·16-s + 16·18-s + 112·22-s + 280·23-s + 142·25-s − 572·29-s − 192·32-s − 48·36-s − 76·37-s − 68·43-s − 336·44-s − 1.12e3·46-s − 568·50-s − 148·53-s + 2.28e3·58-s + 448·64-s + 1.36e3·67-s + 1.17e3·71-s + 128·72-s + 304·74-s + 2.44e3·79-s − 713·81-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s − 0.148·9-s − 0.767·11-s + 5/4·16-s + 0.209·18-s + 1.08·22-s + 2.53·23-s + 1.13·25-s − 3.66·29-s − 1.06·32-s − 2/9·36-s − 0.337·37-s − 0.241·43-s − 1.15·44-s − 3.58·46-s − 1.60·50-s − 0.383·53-s + 5.17·58-s + 7/8·64-s + 2.49·67-s + 1.96·71-s + 0.209·72-s + 0.477·74-s + 3.47·79-s − 0.978·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9604\)    =    \(2^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(33.4336\)
Root analytic conductor: \(2.40461\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{98} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9604,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7872272831\)
\(L(\frac12)\) \(\approx\) \(0.7872272831\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{2} \)
7 \( 1 \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{6} T^{4} \)
5$C_2^2$ \( 1 - 142 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 + 14 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 + 1802 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 + 9824 T^{2} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 13716 T^{2} + p^{6} T^{4} \)
23$C_2$ \( ( 1 - 140 T + p^{3} T^{2} )^{2} \)
29$C_2$ \( ( 1 + 286 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 50870 T^{2} + p^{6} T^{4} \)
37$C_2$ \( ( 1 + 38 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 122000 T^{2} + p^{6} T^{4} \)
43$C_2$ \( ( 1 + 34 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 66154 T^{2} + p^{6} T^{4} \)
53$C_2$ \( ( 1 + 74 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 + 222260 T^{2} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 453762 T^{2} + p^{6} T^{4} \)
67$C_2$ \( ( 1 - 684 T + p^{3} T^{2} )^{2} \)
71$C_2$ \( ( 1 - 588 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 705072 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 - 1220 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 964772 T^{2} + p^{6} T^{4} \)
89$C_2^2$ \( 1 + 1028000 T^{2} + p^{6} T^{4} \)
97$C_2^2$ \( 1 - 375456 T^{2} + p^{6} T^{4} \)
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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

−13.53666154760026102715186056447, −12.95967567290602917403137230869, −12.74715359284140321439404655188, −11.96728387137672545605707381280, −11.14181711493088547971662539773, −10.87355260115927838384521029721, −10.76558519542085117149843533826, −9.577441157677954376548334199689, −9.412279020632818648117706459843, −8.924645867876475589411123580714, −8.103387133035204292888163729514, −7.76394760220724446805700400063, −6.86884639037744538823948784852, −6.77108089612175459885657561003, −5.33526409159283873578687297752, −5.30684781566433497682552596931, −3.71050791864670579925600787107, −2.87528667587869975948663621385, −1.88858810085907637027060777657, −0.64076081636971140382714899099, 0.64076081636971140382714899099, 1.88858810085907637027060777657, 2.87528667587869975948663621385, 3.71050791864670579925600787107, 5.30684781566433497682552596931, 5.33526409159283873578687297752, 6.77108089612175459885657561003, 6.86884639037744538823948784852, 7.76394760220724446805700400063, 8.103387133035204292888163729514, 8.924645867876475589411123580714, 9.412279020632818648117706459843, 9.577441157677954376548334199689, 10.76558519542085117149843533826, 10.87355260115927838384521029721, 11.14181711493088547971662539773, 11.96728387137672545605707381280, 12.74715359284140321439404655188, 12.95967567290602917403137230869, 13.53666154760026102715186056447

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