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Properties

Label 4-69500-1.1-c1e2-0-1
Degree $4$
Conductor $69500$
Sign $1$
Analytic cond. $4.43138$
Root an. cond. $1.45089$
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 − 5-s + 4·9-s − 3·11-s + 16-s − 2·19-s − 20-s + 25-s + 12·29-s + 31-s + 4·36-s − 3·44-s − 4·45-s + 5·49-s + 3·55-s + 12·59-s + 10·61-s + 64-s + 18·71-s − 2·76-s − 11·79-s − 80-s + 7·81-s − 12·89-s + 2·95-s − 12·99-s + 100-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.447·5-s + 4/3·9-s − 0.904·11-s + 1/4·16-s − 0.458·19-s − 0.223·20-s + 1/5·25-s + 2.22·29-s + 0.179·31-s + 2/3·36-s − 0.452·44-s − 0.596·45-s + 5/7·49-s + 0.404·55-s + 1.56·59-s + 1.28·61-s + 1/8·64-s + 2.13·71-s − 0.229·76-s − 1.23·79-s − 0.111·80-s + 7/9·81-s − 1.27·89-s + 0.205·95-s − 1.20·99-s + 1/10·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(69500\)    =    \(2^{2} \cdot 5^{3} \cdot 139\)
Sign: $1$
Analytic conductor: \(4.43138\)
Root analytic conductor: \(1.45089\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 69500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.600620836\)
\(L(\frac12)\) \(\approx\) \(1.600620836\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$ \( 1 + T \)
139$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.3.a_ae
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.7.a_af
11$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.d_w
13$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.13.a_q
17$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.17.a_h
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.19.c_bn
23$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.23.a_bc
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.31.ab_ci
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.41.a_b
43$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.43.a_q
47$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.47.a_af
53$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.53.a_ac
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.59.am_fy
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.61.ak_fi
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.67.a_ack
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.71.as_ig
73$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.73.a_h
79$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.l_gm
83$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.83.a_ach
89$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.89.m_hx
97$C_2^2$ \( 1 - 131 T^{2} + p^{2} T^{4} \) 2.97.a_afb
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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

−10.07559802736835391386159495616, −9.480790150799278218408470513307, −8.663878721382813741480710620516, −8.268284728241120944267470037492, −7.85345654457089869194135514222, −7.22034979433882693891885376766, −6.76173806799556631023758060829, −6.46741724610851946071810440034, −5.50617434765996079219335231984, −5.03862128243239838200856384861, −4.29896063731076874452251228508, −3.87555401289450863122114345380, −2.87296059693570634850062504187, −2.27782502555260193465094698912, −1.08454777609758867210381378945, 1.08454777609758867210381378945, 2.27782502555260193465094698912, 2.87296059693570634850062504187, 3.87555401289450863122114345380, 4.29896063731076874452251228508, 5.03862128243239838200856384861, 5.50617434765996079219335231984, 6.46741724610851946071810440034, 6.76173806799556631023758060829, 7.22034979433882693891885376766, 7.85345654457089869194135514222, 8.268284728241120944267470037492, 8.663878721382813741480710620516, 9.480790150799278218408470513307, 10.07559802736835391386159495616

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