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Properties

Label 4-1274e2-1.1-c1e2-0-27
Degree $4$
Conductor $1623076$
Sign $1$
Analytic cond. $103.488$
Root an. cond. $3.18950$
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  − 2-s − 5-s + 8-s − 6·9-s + 10-s + 8·11-s − 7·13-s − 16-s + 3·17-s + 6·18-s − 8·22-s + 4·23-s + 5·25-s + 7·26-s + 29-s + 4·31-s − 3·34-s − 3·37-s − 40-s − 9·41-s + 8·43-s + 6·45-s − 4·46-s − 8·47-s − 5·50-s + 9·53-s − 8·55-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.447·5-s + 0.353·8-s − 2·9-s + 0.316·10-s + 2.41·11-s − 1.94·13-s − 1/4·16-s + 0.727·17-s + 1.41·18-s − 1.70·22-s + 0.834·23-s + 25-s + 1.37·26-s + 0.185·29-s + 0.718·31-s − 0.514·34-s − 0.493·37-s − 0.158·40-s − 1.40·41-s + 1.21·43-s + 0.894·45-s − 0.589·46-s − 1.16·47-s − 0.707·50-s + 1.23·53-s − 1.07·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1623076 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1623076 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1623076\)    =    \(2^{2} \cdot 7^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(103.488\)
Root analytic conductor: \(3.18950\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1623076,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8999676282\)
\(L(\frac12)\) \(\approx\) \(0.8999676282\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
7 \( 1 \)
13$C_2$ \( 1 + 7 T + p T^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_ae
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.11.ai_bm
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.17.ad_ai
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_ah
29$C_2^2$ \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \) 2.29.ab_abc
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.ae_ap
37$C_2^2$ \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.37.d_abc
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.41.j_bo
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.43.ai_v
47$C_2^2$ \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.47.i_r
53$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.53.aj_bc
59$C_2^2$ \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_abr
61$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.61.o_gp
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2^2$ \( 1 - 8 T - 7 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.71.ai_ah
73$C_2^2$ \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.73.al_bw
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.79.ae_acl
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.89.g_acb
97$C_2^2$ \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.97.ac_adp
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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

−9.584072725956746237973349572380, −9.351964053554124795378698358521, −9.242088651756470189714322422692, −8.656188529841566323740662880622, −8.242106424307843831340100725985, −8.147632836067489378688335317932, −7.43030305762522662513686648369, −7.02572093811939281490106874204, −6.62639983009215043486445830464, −6.40724720025065079162952937128, −5.51120108154346794493574770742, −5.43322216959255624793171134783, −4.59629068514213009223124216308, −4.53743838747959346643601269595, −3.54023507096519843056312207290, −3.32862262084574237466263846843, −2.72606983128633489630764405152, −2.09923813210027357991416000185, −1.20503749148774732891790408622, −0.50605443589789670863271595371, 0.50605443589789670863271595371, 1.20503749148774732891790408622, 2.09923813210027357991416000185, 2.72606983128633489630764405152, 3.32862262084574237466263846843, 3.54023507096519843056312207290, 4.53743838747959346643601269595, 4.59629068514213009223124216308, 5.43322216959255624793171134783, 5.51120108154346794493574770742, 6.40724720025065079162952937128, 6.62639983009215043486445830464, 7.02572093811939281490106874204, 7.43030305762522662513686648369, 8.147632836067489378688335317932, 8.242106424307843831340100725985, 8.656188529841566323740662880622, 9.242088651756470189714322422692, 9.351964053554124795378698358521, 9.584072725956746237973349572380

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