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Properties

Label 4-148e2-1.1-c1e2-0-10
Degree $4$
Conductor $21904$
Sign $1$
Analytic cond. $1.39661$
Root an. cond. $1.08709$
Motivic weight $1$
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  + 2·2-s + 2·4-s + 6·5-s − 6·9-s + 12·10-s − 10·13-s − 4·16-s + 6·17-s − 12·18-s + 12·20-s + 18·25-s − 20·26-s − 6·29-s − 8·32-s + 12·34-s − 12·36-s + 12·37-s − 36·45-s + 14·49-s + 36·50-s − 20·52-s + 8·53-s − 12·58-s − 22·61-s − 8·64-s − 60·65-s + 12·68-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 2.68·5-s − 2·9-s + 3.79·10-s − 2.77·13-s − 16-s + 1.45·17-s − 2.82·18-s + 2.68·20-s + 18/5·25-s − 3.92·26-s − 1.11·29-s − 1.41·32-s + 2.05·34-s − 2·36-s + 1.97·37-s − 5.36·45-s + 2·49-s + 5.09·50-s − 2.77·52-s + 1.09·53-s − 1.57·58-s − 2.81·61-s − 64-s − 7.44·65-s + 1.45·68-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21904\)    =    \(2^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(1.39661\)
Root analytic conductor: \(1.08709\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21904,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.787622809\)
\(L(\frac12)\) \(\approx\) \(2.787622809\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2$C_2$ \( 1 - p T + p T^{2} \)
37$C_2$ \( 1 - 12 T + p T^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.5.ag_s
7$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.7.a_ao
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.k_by
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.ag_s
19$C_2^2$ \( 1 + p^{2} T^{4} \) 2.19.a_a
23$C_2^2$ \( 1 + p^{2} T^{4} \) 2.23.a_a
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.g_s
31$C_2^2$ \( 1 + p^{2} T^{4} \) 2.31.a_a
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.a_s
43$C_2^2$ \( 1 + p^{2} T^{4} \) 2.43.a_a
47$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.47.a_adq
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.53.ai_es
59$C_2^2$ \( 1 + p^{2} T^{4} \) 2.59.a_a
61$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.w_ji
67$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.67.a_fe
71$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.71.a_afm
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.a_aeg
79$C_2^2$ \( 1 + p^{2} T^{4} \) 2.79.a_a
83$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.83.a_agk
89$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) 2.89.aba_na
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.97.ak_by
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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

−13.38295322907736787778881326635, −13.00990653223404041069085592191, −12.33354705228281346665432087070, −11.98079613566299152672880204585, −11.64963749869419349558325972927, −10.66599652764054965452945764723, −10.32910330103679956927225347313, −9.624309784556378630629331724724, −9.262893026468890336311563424058, −9.080929980070047352098562688908, −7.85446052775446857041677386812, −7.36648869783280391306701732102, −6.38307137340059184778249816148, −5.96433320584729864389058438387, −5.42815115634598917646662852045, −5.39396683365954570854227555283, −4.60074539459330830700170132501, −3.22613434108872948136651348365, −2.42587430620713204501231134824, −2.35351817328274456471123048989, 2.35351817328274456471123048989, 2.42587430620713204501231134824, 3.22613434108872948136651348365, 4.60074539459330830700170132501, 5.39396683365954570854227555283, 5.42815115634598917646662852045, 5.96433320584729864389058438387, 6.38307137340059184778249816148, 7.36648869783280391306701732102, 7.85446052775446857041677386812, 9.080929980070047352098562688908, 9.262893026468890336311563424058, 9.624309784556378630629331724724, 10.32910330103679956927225347313, 10.66599652764054965452945764723, 11.64963749869419349558325972927, 11.98079613566299152672880204585, 12.33354705228281346665432087070, 13.00990653223404041069085592191, 13.38295322907736787778881326635

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