Properties

Label 4-98e2-1.1-c1e2-0-1
Degree $4$
Conductor $9604$
Sign $1$
Analytic cond. $0.612359$
Root an. cond. $0.884609$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s + 4·8-s − 4·9-s − 4·11-s + 5·16-s − 8·18-s − 8·22-s − 8·23-s − 2·25-s + 4·29-s + 6·32-s − 12·36-s + 20·37-s + 4·43-s − 12·44-s − 16·46-s − 4·50-s − 4·53-s + 8·58-s + 7·64-s + 24·67-s − 24·71-s − 16·72-s + 40·74-s − 8·79-s + 7·81-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s − 4/3·9-s − 1.20·11-s + 5/4·16-s − 1.88·18-s − 1.70·22-s − 1.66·23-s − 2/5·25-s + 0.742·29-s + 1.06·32-s − 2·36-s + 3.28·37-s + 0.609·43-s − 1.80·44-s − 2.35·46-s − 0.565·50-s − 0.549·53-s + 1.05·58-s + 7/8·64-s + 2.93·67-s − 2.84·71-s − 1.88·72-s + 4.64·74-s − 0.900·79-s + 7/9·81-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(9604\)    =    \(2^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.612359\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{98} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9604,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.896963851\)
\(L(\frac12)\) \(\approx\) \(1.896963851\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 12 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 116 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 144 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 128 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 96 T^{2} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.08889201258632171177809522471, −13.75897892764682158104944478298, −12.99181020642601878454218038210, −12.96065310228585205937755138639, −11.91815093620013056118481872743, −11.88110889862418557601494564222, −11.08749216798728719416053267125, −10.79739703594736369657619807538, −9.981854834006967455041537620725, −9.508258444468192616988098371622, −8.371218444153554891246935980326, −8.041979252561494532787930581368, −7.48845653750310791794370134824, −6.48255113005534387242997547576, −5.86451827160416506222883333068, −5.60287084867711068938249624170, −4.66415541315340900968485969565, −4.02076711545202908644935151541, −2.90852999167922493658293827195, −2.41717302449154006608420706146, 2.41717302449154006608420706146, 2.90852999167922493658293827195, 4.02076711545202908644935151541, 4.66415541315340900968485969565, 5.60287084867711068938249624170, 5.86451827160416506222883333068, 6.48255113005534387242997547576, 7.48845653750310791794370134824, 8.041979252561494532787930581368, 8.371218444153554891246935980326, 9.508258444468192616988098371622, 9.981854834006967455041537620725, 10.79739703594736369657619807538, 11.08749216798728719416053267125, 11.88110889862418557601494564222, 11.91815093620013056118481872743, 12.96065310228585205937755138639, 12.99181020642601878454218038210, 13.75897892764682158104944478298, 14.08889201258632171177809522471

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