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Properties

Label 4-837e2-1.1-c1e2-0-6
Degree $4$
Conductor $700569$
Sign $1$
Analytic cond. $44.6688$
Root an. cond. $2.58524$
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  + 4-s + 4·7-s − 3·16-s − 6·17-s + 8·19-s − 12·23-s − 2·25-s + 4·28-s + 6·29-s + 4·31-s + 7·49-s + 12·53-s − 7·64-s + 8·67-s − 6·68-s + 8·76-s + 24·83-s − 6·89-s − 12·92-s + 4·97-s − 2·100-s + 20·103-s + 10·109-s − 12·112-s + 6·116-s − 24·119-s − 13·121-s + ⋯
L(s)  = 1  + 1/2·4-s + 1.51·7-s − 3/4·16-s − 1.45·17-s + 1.83·19-s − 2.50·23-s − 2/5·25-s + 0.755·28-s + 1.11·29-s + 0.718·31-s + 49-s + 1.64·53-s − 7/8·64-s + 0.977·67-s − 0.727·68-s + 0.917·76-s + 2.63·83-s − 0.635·89-s − 1.25·92-s + 0.406·97-s − 1/5·100-s + 1.97·103-s + 0.957·109-s − 1.13·112-s + 0.557·116-s − 2.20·119-s − 1.18·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(700569\)    =    \(3^{6} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(44.6688\)
Root analytic conductor: \(2.58524\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 700569,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.558377323\)
\(L(\frac12)\) \(\approx\) \(2.558377323\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3 \( 1 \)
31$C_2$ \( 1 - 4 T + p T^{2} \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.2.a_ab
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.ae_j
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.a_n
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.17.g_br
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.19.ai_bt
23$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.23.m_cv
29$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.29.ag_bf
37$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.37.a_bi
41$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.41.a_c
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.a_ao
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.59.a_acs
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.a_cg
67$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.67.ai_cr
71$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.71.a_o
73$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.73.a_afq
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.79.a_aby
83$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) 2.83.ay_lp
89$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.89.g_hf
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.ae_cc
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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

−8.269603903303497757677889855345, −7.84000908173030278163812418806, −7.53694584662574347477053711043, −7.04942140074371120603101087477, −6.42944584647231090846166238004, −6.21754171652954419491713784023, −5.50456583930453353689226122455, −5.09289525253428726295898867338, −4.55478515362523927269779923992, −4.21048679372950353673855608623, −3.60808837109263522919091099914, −2.75385958119311448430490806839, −2.11634734949427852296621665680, −1.86566528791837902527141286186, −0.791780233331230027299848113700, 0.791780233331230027299848113700, 1.86566528791837902527141286186, 2.11634734949427852296621665680, 2.75385958119311448430490806839, 3.60808837109263522919091099914, 4.21048679372950353673855608623, 4.55478515362523927269779923992, 5.09289525253428726295898867338, 5.50456583930453353689226122455, 6.21754171652954419491713784023, 6.42944584647231090846166238004, 7.04942140074371120603101087477, 7.53694584662574347477053711043, 7.84000908173030278163812418806, 8.269603903303497757677889855345

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