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Properties

Label 4-364e2-1.1-c1e2-0-12
Degree $4$
Conductor $132496$
Sign $1$
Analytic cond. $8.44805$
Root an. cond. $1.70486$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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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 − 2·13-s − 4·16-s + 8·17-s + 12·18-s − 12·20-s + 17·25-s + 4·26-s − 10·29-s + 8·32-s − 16·34-s − 12·36-s − 8·37-s − 12·41-s + 36·45-s + 49-s − 34·50-s − 4·52-s − 18·53-s + 20·58-s − 20·61-s − 8·64-s + 12·65-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 2.68·5-s − 2·9-s + 3.79·10-s − 0.554·13-s − 16-s + 1.94·17-s + 2.82·18-s − 2.68·20-s + 17/5·25-s + 0.784·26-s − 1.85·29-s + 1.41·32-s − 2.74·34-s − 2·36-s − 1.31·37-s − 1.87·41-s + 5.36·45-s + 1/7·49-s − 4.80·50-s − 0.554·52-s − 2.47·53-s + 2.62·58-s − 2.56·61-s − 64-s + 1.48·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(132496\)    =    \(2^{4} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(8.44805\)
Root analytic conductor: \(1.70486\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 132496,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + p T + p T^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.5.g_t
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.a_ao
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.17.ai_by
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.19.a_n
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.a_bl
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.29.k_df
31$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.31.a_cb
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.37.i_dm
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.43.a_dh
47$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.47.a_bt
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.53.s_hf
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.a_cc
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.67.a_du
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.a_da
73$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.73.ba_md
79$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.79.a_ft
83$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.83.a_ach
89$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.89.ag_hf
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.97.ao_jj
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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

−8.920719672465111957084037828236, −8.306653899443735547493881784402, −7.88305646625584781075065151406, −7.65856427227425769928738953272, −7.43563929744309328324545511818, −6.69324481208496363874135514467, −5.91107970337180073579003850586, −5.27883313770925671699270459709, −4.71772169836810783436044308640, −3.91316951599567757387300319071, −3.20361346457630305584783093767, −3.11744276397134147631028514628, −1.62321334030766970947302300597, 0, 0, 1.62321334030766970947302300597, 3.11744276397134147631028514628, 3.20361346457630305584783093767, 3.91316951599567757387300319071, 4.71772169836810783436044308640, 5.27883313770925671699270459709, 5.91107970337180073579003850586, 6.69324481208496363874135514467, 7.43563929744309328324545511818, 7.65856427227425769928738953272, 7.88305646625584781075065151406, 8.306653899443735547493881784402, 8.920719672465111957084037828236

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