Properties

Label 4-135200-1.1-c1e2-0-15
Degree $4$
Conductor $135200$
Sign $-1$
Analytic cond. $8.62046$
Root an. cond. $1.71349$
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-s + 4-s − 8-s − 4·9-s + 4·13-s + 16-s − 6·17-s + 4·18-s − 5·25-s − 4·26-s + 6·29-s − 32-s + 6·34-s − 4·36-s + 4·37-s − 12·41-s + 14·49-s + 5·50-s + 4·52-s − 12·53-s − 6·58-s + 2·61-s + 64-s − 6·68-s + 4·72-s + 4·73-s − 4·74-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 4/3·9-s + 1.10·13-s + 1/4·16-s − 1.45·17-s + 0.942·18-s − 25-s − 0.784·26-s + 1.11·29-s − 0.176·32-s + 1.02·34-s − 2/3·36-s + 0.657·37-s − 1.87·41-s + 2·49-s + 0.707·50-s + 0.554·52-s − 1.64·53-s − 0.787·58-s + 0.256·61-s + 1/8·64-s − 0.727·68-s + 0.471·72-s + 0.468·73-s − 0.464·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(135200\)    =    \(2^{5} \cdot 5^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(8.62046\)
Root analytic conductor: \(1.71349\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 135200,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
5$C_2$ \( 1 + p T^{2} \)
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$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.7.a_ao
11$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.11.a_ae
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.g_bi
19$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.19.a_aq
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.29.ag_cg
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.31.a_ba
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2^2$ \( 1 - 76 T^{2} + p^{2} T^{4} \) 2.43.a_acy
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \) 2.59.a_dc
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.ac_bq
67$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.67.a_aba
71$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.71.a_aec
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.79.a_ao
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.83.a_cg
89$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.s_jq
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.97.q_jy
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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

−8.924295615658453126135095443866, −8.623350911651288595526567333958, −8.325472870543069950434464805336, −7.88509588219189576350649687242, −7.13213035408466851250540136520, −6.56250211478893505456282074100, −6.28359205066225115557299358498, −5.67089076521895748696194893885, −5.17632291403142334999749550722, −4.30868469828010056206803814752, −3.75840718598280598900188342871, −2.91729748648867155400421287788, −2.41198268533464804198389187296, −1.41750518557827308865575572420, 0, 1.41750518557827308865575572420, 2.41198268533464804198389187296, 2.91729748648867155400421287788, 3.75840718598280598900188342871, 4.30868469828010056206803814752, 5.17632291403142334999749550722, 5.67089076521895748696194893885, 6.28359205066225115557299358498, 6.56250211478893505456282074100, 7.13213035408466851250540136520, 7.88509588219189576350649687242, 8.325472870543069950434464805336, 8.623350911651288595526567333958, 8.924295615658453126135095443866

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