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Properties

Label 4-792e2-1.1-c1e2-0-126
Degree $4$
Conductor $627264$
Sign $-1$
Analytic cond. $39.9948$
Root an. cond. $2.51478$
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  + 2·2-s + 2·4-s + 2·11-s − 4·16-s + 2·17-s + 4·22-s + 3·25-s − 8·29-s − 14·31-s − 8·32-s + 4·34-s − 22·41-s + 4·44-s − 6·49-s + 6·50-s − 16·58-s − 28·62-s − 8·64-s − 12·67-s + 4·68-s − 44·82-s + 12·83-s − 10·97-s − 12·98-s + 6·100-s + 2·101-s + 18·107-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 0.603·11-s − 16-s + 0.485·17-s + 0.852·22-s + 3/5·25-s − 1.48·29-s − 2.51·31-s − 1.41·32-s + 0.685·34-s − 3.43·41-s + 0.603·44-s − 6/7·49-s + 0.848·50-s − 2.10·58-s − 3.55·62-s − 64-s − 1.46·67-s + 0.485·68-s − 4.85·82-s + 1.31·83-s − 1.01·97-s − 1.21·98-s + 3/5·100-s + 0.199·101-s + 1.74·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 627264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(627264\)    =    \(2^{6} \cdot 3^{4} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(39.9948\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 627264,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - p T + p T^{2} \)
3 \( 1 \)
11$C_2$ \( 1 - 2 T + p T^{2} \)
good5$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.5.a_ad
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
13$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.13.a_as
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.ac_k
19$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.19.a_ag
23$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \) 2.23.a_aj
29$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.i_cs
31$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.31.o_ed
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.37.a_cn
41$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.w_hu
43$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.43.a_abe
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.53.a_abe
59$C_2^2$ \( 1 + 81 T^{2} + p^{2} T^{4} \) 2.59.a_dd
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.67.m_gf
71$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.71.a_l
73$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.73.a_acc
79$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.79.a_s
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.83.am_hu
89$C_2^2$ \( 1 + 27 T^{2} + p^{2} T^{4} \) 2.89.a_bb
97$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.97.k_hv
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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

−8.049482569072556957084623899644, −7.56615086280161211634482849681, −7.02380893687740458028342182937, −6.77967599288021445811070982425, −6.25192664355428406770711078675, −5.64604133292153262797010919189, −5.39211491249495973511722312830, −4.92722060787361327282561211012, −4.41192491011343815654726284063, −3.65700393671907007804811149800, −3.52584436216380051991723342939, −3.02394747843726523989697691673, −1.98977316231956606937571008941, −1.62807662458849928650119019247, 0, 1.62807662458849928650119019247, 1.98977316231956606937571008941, 3.02394747843726523989697691673, 3.52584436216380051991723342939, 3.65700393671907007804811149800, 4.41192491011343815654726284063, 4.92722060787361327282561211012, 5.39211491249495973511722312830, 5.64604133292153262797010919189, 6.25192664355428406770711078675, 6.77967599288021445811070982425, 7.02380893687740458028342182937, 7.56615086280161211634482849681, 8.049482569072556957084623899644

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