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Properties

Label 4-832e2-1.1-c1e2-0-14
Degree $4$
Conductor $692224$
Sign $1$
Analytic cond. $44.1368$
Root an. cond. $2.57750$
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  − 2·5-s + 6·9-s − 4·13-s + 2·25-s + 8·29-s + 14·37-s − 18·41-s − 12·45-s + 28·53-s − 20·61-s + 8·65-s + 10·73-s + 27·81-s − 26·89-s + 26·97-s − 26·109-s + 32·113-s − 24·117-s − 10·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 0.894·5-s + 2·9-s − 1.10·13-s + 2/5·25-s + 1.48·29-s + 2.30·37-s − 2.81·41-s − 1.78·45-s + 3.84·53-s − 2.56·61-s + 0.992·65-s + 1.17·73-s + 3·81-s − 2.75·89-s + 2.63·97-s − 2.49·109-s + 3.01·113-s − 2.21·117-s − 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.32·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 692224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 692224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(692224\)    =    \(2^{12} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(44.1368\)
Root analytic conductor: \(2.57750\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 692224,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.810375262\)
\(L(\frac12)\) \(\approx\) \(1.810375262\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.3.a_ag
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.c_c
7$C_2^2$ \( 1 + p^{2} T^{4} \) 2.7.a_a
11$C_2^2$ \( 1 + p^{2} T^{4} \) 2.11.a_a
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.a_abe
19$C_2^2$ \( 1 + p^{2} T^{4} \) 2.19.a_a
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.29.ai_cw
31$C_2^2$ \( 1 + p^{2} T^{4} \) 2.31.a_a
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.37.ao_du
41$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.s_gg
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2^2$ \( 1 + p^{2} T^{4} \) 2.47.a_a
53$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.53.abc_lq
59$C_2^2$ \( 1 + p^{2} T^{4} \) 2.59.a_a
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2^2$ \( 1 + p^{2} T^{4} \) 2.67.a_a
71$C_2^2$ \( 1 + p^{2} T^{4} \) 2.71.a_a
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.ak_by
79$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.79.a_agc
83$C_2^2$ \( 1 + p^{2} T^{4} \) 2.83.a_a
89$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.89.ba_na
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

−10.25373964146720309162317462401, −10.11081235931218808484775954777, −9.723464124329590588067082061597, −9.221472717260353864888676080485, −8.675822134031912651049438441443, −8.278059953066293564914083222283, −7.69919666610636119465184847023, −7.51141104987534234324747991228, −6.99816290123132793083542187750, −6.74123374497659023336279004524, −6.23347567113737762826082679742, −5.47727154638230615836880122122, −4.83401247899382478262749333376, −4.63288123618797833350163336783, −4.10628128650544631600482402891, −3.71539162375779761475972452083, −2.93339273192081698933226158224, −2.34251563079843641397516445589, −1.51604154383517791386100761892, −0.71060958710711567148670955373, 0.71060958710711567148670955373, 1.51604154383517791386100761892, 2.34251563079843641397516445589, 2.93339273192081698933226158224, 3.71539162375779761475972452083, 4.10628128650544631600482402891, 4.63288123618797833350163336783, 4.83401247899382478262749333376, 5.47727154638230615836880122122, 6.23347567113737762826082679742, 6.74123374497659023336279004524, 6.99816290123132793083542187750, 7.51141104987534234324747991228, 7.69919666610636119465184847023, 8.278059953066293564914083222283, 8.675822134031912651049438441443, 9.221472717260353864888676080485, 9.723464124329590588067082061597, 10.11081235931218808484775954777, 10.25373964146720309162317462401

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