Properties

Label 4-216320-1.1-c1e2-0-0
Degree $4$
Conductor $216320$
Sign $1$
Analytic cond. $13.7927$
Root an. cond. $1.92713$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s − 4·9-s + 4·13-s + 2·25-s + 2·29-s − 4·37-s − 8·41-s + 12·45-s + 10·49-s + 2·53-s − 10·61-s − 12·65-s + 7·81-s + 4·89-s + 26·97-s + 20·101-s − 14·109-s + 2·113-s − 16·117-s − 10·121-s + 10·125-s + ⋯
L(s)  = 1  − 1.34·5-s − 4/3·9-s + 1.10·13-s + 2/5·25-s + 0.371·29-s − 0.657·37-s − 1.24·41-s + 1.78·45-s + 10/7·49-s + 0.274·53-s − 1.28·61-s − 1.48·65-s + 7/9·81-s + 0.423·89-s + 2.63·97-s + 1.99·101-s − 1.34·109-s + 0.188·113-s − 1.47·117-s − 0.909·121-s + 0.894·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216320 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216320 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(216320\)    =    \(2^{8} \cdot 5 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(13.7927\)
Root analytic conductor: \(1.92713\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 216320,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8914772799\)
\(L(\frac12)\) \(\approx\) \(0.8914772799\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( 1 - 4 T + p T^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.23.a_ba
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.ac_bi
31$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.31.a_au
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.i_ck
43$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.43.a_q
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.ac_de
59$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.59.a_acs
61$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.k_es
67$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.67.a_aby
71$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.71.a_bo
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.a_eg
79$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.79.a_agc
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \) 2.83.a_dm
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.ae_eo
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.97.aba_na
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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

−8.895546895401547173695072881709, −8.500199784705780800277410143132, −8.227730545593475736026147109062, −7.64843380273346974031086293073, −7.28229832882360434549394096249, −6.60699000115188559694116158710, −6.12178356948425297205005149366, −5.66649998447074351537305636940, −5.05157594366465098526615991399, −4.45378788525558738424735133000, −3.78190437992248957968170737493, −3.40716085167138754556436142269, −2.84570656103374590911141062145, −1.85681222093042714789929357226, −0.58553341626313278577283406574, 0.58553341626313278577283406574, 1.85681222093042714789929357226, 2.84570656103374590911141062145, 3.40716085167138754556436142269, 3.78190437992248957968170737493, 4.45378788525558738424735133000, 5.05157594366465098526615991399, 5.66649998447074351537305636940, 6.12178356948425297205005149366, 6.60699000115188559694116158710, 7.28229832882360434549394096249, 7.64843380273346974031086293073, 8.227730545593475736026147109062, 8.500199784705780800277410143132, 8.895546895401547173695072881709

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