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Properties

Label 4-1218e2-1.1-c1e2-0-17
Degree $4$
Conductor $1483524$
Sign $-1$
Analytic cond. $94.5907$
Root an. cond. $3.11861$
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  + 4-s − 9-s − 2·11-s + 16-s + 8·23-s − 2·25-s − 36-s − 10·37-s − 2·44-s − 7·49-s − 10·53-s + 64-s + 14·67-s + 4·71-s + 2·79-s + 81-s + 8·92-s + 2·99-s − 2·100-s − 26·107-s − 2·109-s + 8·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 1/2·4-s − 1/3·9-s − 0.603·11-s + 1/4·16-s + 1.66·23-s − 2/5·25-s − 1/6·36-s − 1.64·37-s − 0.301·44-s − 49-s − 1.37·53-s + 1/8·64-s + 1.71·67-s + 0.474·71-s + 0.225·79-s + 1/9·81-s + 0.834·92-s + 0.201·99-s − 1/5·100-s − 2.51·107-s − 0.191·109-s + 0.752·113-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1483524 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1483524 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1483524\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 29^{2}\)
Sign: $-1$
Analytic conductor: \(94.5907\)
Root analytic conductor: \(3.11861\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1483524,\ (\ :1/2, 1/2),\ -1)\)

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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_2$ \( 1 + T^{2} \)
7$C_2$ \( 1 + p T^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.c_o
13$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.13.a_c
17$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.17.a_i
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.23.ai_ck
31$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.31.a_bo
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.k_by
41$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.41.a_bo
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \) 2.47.a_cm
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.k_de
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.59.a_s
61$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.61.a_ec
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.67.ao_gc
71$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.ae_de
73$C_2^2$ \( 1 - 104 T^{2} + p^{2} T^{4} \) 2.73.a_aea
79$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.ac_da
83$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.83.a_aco
89$C_2^2$ \( 1 - 100 T^{2} + p^{2} T^{4} \) 2.89.a_adw
97$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.97.a_bc
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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

−7.68214348945521665916401469614, −7.23422313828616334813638422080, −6.84939523221067205527702879173, −6.45639595477600428810592124085, −6.03354168409305844038308864277, −5.40358938874246969991449875839, −5.03907078874390501264094283337, −4.83506159829097937478939903350, −3.92648667678561485306490432515, −3.50837371704112427027043001312, −2.97725197833586707510778940428, −2.51790214174220780720121214879, −1.84211043974220819449786196350, −1.14352855352875573987549638715, 0, 1.14352855352875573987549638715, 1.84211043974220819449786196350, 2.51790214174220780720121214879, 2.97725197833586707510778940428, 3.50837371704112427027043001312, 3.92648667678561485306490432515, 4.83506159829097937478939903350, 5.03907078874390501264094283337, 5.40358938874246969991449875839, 6.03354168409305844038308864277, 6.45639595477600428810592124085, 6.84939523221067205527702879173, 7.23422313828616334813638422080, 7.68214348945521665916401469614

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