Properties

Label 4-416000-1.1-c1e2-0-13
Degree $4$
Conductor $416000$
Sign $-1$
Analytic cond. $26.5245$
Root an. cond. $2.26940$
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  − 5-s − 2·9-s − 5·13-s + 8·17-s + 25-s + 4·29-s − 16·37-s + 12·41-s + 2·45-s − 2·49-s − 16·53-s + 12·61-s + 5·65-s − 16·73-s − 5·81-s − 8·85-s − 20·89-s + 12·101-s + 4·109-s + 16·113-s + 10·117-s − 14·121-s − 125-s + ⋯
L(s)  = 1  − 0.447·5-s − 2/3·9-s − 1.38·13-s + 1.94·17-s + 1/5·25-s + 0.742·29-s − 2.63·37-s + 1.87·41-s + 0.298·45-s − 2/7·49-s − 2.19·53-s + 1.53·61-s + 0.620·65-s − 1.87·73-s − 5/9·81-s − 0.867·85-s − 2.11·89-s + 1.19·101-s + 0.383·109-s + 1.50·113-s + 0.924·117-s − 1.27·121-s − 0.0894·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(416000\)    =    \(2^{8} \cdot 5^{3} \cdot 13\)
Sign: $-1$
Analytic conductor: \(26.5245\)
Root analytic conductor: \(2.26940\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 416000,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
5$C_1$ \( 1 + T \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 6 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.17.ai_bu
19$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.19.a_o
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.23.a_c
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.ae_bu
31$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.31.a_abi
37$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.q_fe
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.43.a_s
47$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.47.a_s
53$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.q_gk
59$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.59.a_be
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.61.am_fm
67$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \) 2.67.a_es
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.71.a_afa
73$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.q_hy
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.a_adu
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.u_kc
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.a_dq
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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

−8.353882642144785999610963741383, −7.918471281969388939658164535715, −7.47613531413283927176346611462, −7.17370418419080124435682553727, −6.62232668618412699421431579051, −5.94851523567293533508956785486, −5.53279241065322665314242727099, −5.09727663943199687493674172191, −4.60532836888284591256651865411, −3.96598494475268625993234644019, −3.15406494069201593720255864670, −3.04023382254879918412243438763, −2.14712279848930932802358658081, −1.20505066052664106418192159720, 0, 1.20505066052664106418192159720, 2.14712279848930932802358658081, 3.04023382254879918412243438763, 3.15406494069201593720255864670, 3.96598494475268625993234644019, 4.60532836888284591256651865411, 5.09727663943199687493674172191, 5.53279241065322665314242727099, 5.94851523567293533508956785486, 6.62232668618412699421431579051, 7.17370418419080124435682553727, 7.47613531413283927176346611462, 7.918471281969388939658164535715, 8.353882642144785999610963741383

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