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Properties

Label 4-987696-1.1-c1e2-0-10
Degree $4$
Conductor $987696$
Sign $-1$
Analytic cond. $62.9763$
Root an. cond. $2.81704$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s − 4-s + 5·5-s + 2·6-s + 3·8-s + 3·9-s − 5·10-s + 2·12-s − 10·15-s − 16-s − 3·17-s − 3·18-s + 19-s − 5·20-s − 6·24-s + 11·25-s − 4·27-s + 10·30-s − 7·31-s − 5·32-s + 3·34-s − 3·36-s − 38-s + 15·40-s + 15·45-s + 2·48-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 1/2·4-s + 2.23·5-s + 0.816·6-s + 1.06·8-s + 9-s − 1.58·10-s + 0.577·12-s − 2.58·15-s − 1/4·16-s − 0.727·17-s − 0.707·18-s + 0.229·19-s − 1.11·20-s − 1.22·24-s + 11/5·25-s − 0.769·27-s + 1.82·30-s − 1.25·31-s − 0.883·32-s + 0.514·34-s − 1/2·36-s − 0.162·38-s + 2.37·40-s + 2.23·45-s + 0.288·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(987696\)    =    \(2^{4} \cdot 3^{2} \cdot 19^{3}\)
Sign: $-1$
Analytic conductor: \(62.9763\)
Root analytic conductor: \(2.81704\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 987696,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + p T^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
19$C_1$ \( 1 - T \)
good5$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.5.af_o
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.a_ao
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.13.a_i
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
23$C_2^2$ \( 1 - 24 T^{2} + p^{2} T^{4} \) 2.23.a_ay
29$C_2^2$ \( 1 + 27 T^{2} + p^{2} T^{4} \) 2.29.a_bb
31$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.h_cc
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.a_k
41$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.41.a_ad
43$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.43.a_z
47$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \) 2.47.a_acq
53$C_2^2$ \( 1 + 79 T^{2} + p^{2} T^{4} \) 2.53.a_db
59$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.59.m_fp
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.61.af_cu
67$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.67.b_fc
71$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.j_ec
73$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.73.ac_d
79$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.79.a_o
83$C_2^2$ \( 1 - 72 T^{2} + p^{2} T^{4} \) 2.83.a_acu
89$C_2^2$ \( 1 + 165 T^{2} + p^{2} T^{4} \) 2.89.a_gj
97$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.97.a_cg
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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

−7.81947458032215673314917532689, −7.49134181637822409019182211683, −6.90820799466782018221444454712, −6.51538629427949879427875269097, −6.08900451382462637094878921273, −5.66001524938879741637465838582, −5.36968156418594919959726385528, −4.88323745463589915070432771723, −4.46103397436716246743238818421, −3.82221976196370896553630440730, −2.99753924749007578976059399541, −2.09439797072401209883470616174, −1.76289656293079236896734150247, −1.12946967656516647051616965624, 0, 1.12946967656516647051616965624, 1.76289656293079236896734150247, 2.09439797072401209883470616174, 2.99753924749007578976059399541, 3.82221976196370896553630440730, 4.46103397436716246743238818421, 4.88323745463589915070432771723, 5.36968156418594919959726385528, 5.66001524938879741637465838582, 6.08900451382462637094878921273, 6.51538629427949879427875269097, 6.90820799466782018221444454712, 7.49134181637822409019182211683, 7.81947458032215673314917532689

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