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Properties

Label 4-4650e2-1.1-c1e2-0-23
Degree $4$
Conductor $21622500$
Sign $1$
Analytic cond. $1378.66$
Root an. cond. $6.09347$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 9-s + 10·11-s + 16-s − 2·19-s − 8·29-s − 2·31-s + 36-s + 8·41-s − 10·44-s + 13·49-s + 20·59-s + 20·61-s − 64-s + 30·71-s + 2·76-s − 26·79-s + 81-s − 6·89-s − 10·99-s + 6·101-s + 8·116-s + 53·121-s + 2·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s + 3.01·11-s + 1/4·16-s − 0.458·19-s − 1.48·29-s − 0.359·31-s + 1/6·36-s + 1.24·41-s − 1.50·44-s + 13/7·49-s + 2.60·59-s + 2.56·61-s − 1/8·64-s + 3.56·71-s + 0.229·76-s − 2.92·79-s + 1/9·81-s − 0.635·89-s − 1.00·99-s + 0.597·101-s + 0.742·116-s + 4.81·121-s + 0.179·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21622500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21622500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21622500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(1378.66\)
Root analytic conductor: \(6.09347\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21622500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.376910328\)
\(L(\frac12)\) \(\approx\) \(3.376910328\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
31$C_1$ \( ( 1 + T )^{2} \)
good7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.7.a_an
11$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.11.ak_bv
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.19.c_bn
23$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \) 2.23.a_av
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.29.i_cw
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.41.ai_du
43$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \) 2.43.a_bj
47$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.47.a_g
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.53.a_az
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.59.au_ik
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.61.au_io
67$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \) 2.67.a_adu
71$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \) 2.71.abe_od
73$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \) 2.73.a_x
79$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.79.ba_mp
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.83.a_ady
89$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.89.g_hf
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.97.a_fa
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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

−8.615699458092046516275293476461, −8.423572104079120351787870354213, −7.79116333549831210881321658708, −7.13942787609982041771245527436, −7.03522856796356124407241037527, −6.87425895314075708694153450601, −6.27548702731273167083856938138, −5.86554616156366003064539754777, −5.69194680224479192282528888336, −5.26621339764228355092942823526, −4.70075747341214361766794937972, −4.08937511018525118454945297801, −4.06411247931206609341723310623, −3.65826437405390530557472790923, −3.44682457929482028615804024730, −2.46699905022219691888883915934, −2.22812170323160894937703197808, −1.61159020013082023871233493669, −0.978594959511114111820810843533, −0.63513392423936788866804755703, 0.63513392423936788866804755703, 0.978594959511114111820810843533, 1.61159020013082023871233493669, 2.22812170323160894937703197808, 2.46699905022219691888883915934, 3.44682457929482028615804024730, 3.65826437405390530557472790923, 4.06411247931206609341723310623, 4.08937511018525118454945297801, 4.70075747341214361766794937972, 5.26621339764228355092942823526, 5.69194680224479192282528888336, 5.86554616156366003064539754777, 6.27548702731273167083856938138, 6.87425895314075708694153450601, 7.03522856796356124407241037527, 7.13942787609982041771245527436, 7.79116333549831210881321658708, 8.423572104079120351787870354213, 8.615699458092046516275293476461

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