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Properties

Label 4-462e2-1.1-c1e2-0-6
Degree $4$
Conductor $213444$
Sign $1$
Analytic cond. $13.6093$
Root an. cond. $1.92070$
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 + 9-s − 4·11-s + 16-s − 2·25-s + 4·28-s + 16·29-s + 36-s − 12·37-s − 16·43-s − 4·44-s + 9·49-s + 20·53-s + 4·63-s + 64-s + 8·67-s + 24·71-s − 16·77-s + 8·79-s + 81-s − 4·99-s − 2·100-s + 16·107-s + 8·109-s + 4·112-s − 4·113-s + ⋯
L(s)  = 1  + 1/2·4-s + 1.51·7-s + 1/3·9-s − 1.20·11-s + 1/4·16-s − 2/5·25-s + 0.755·28-s + 2.97·29-s + 1/6·36-s − 1.97·37-s − 2.43·43-s − 0.603·44-s + 9/7·49-s + 2.74·53-s + 0.503·63-s + 1/8·64-s + 0.977·67-s + 2.84·71-s − 1.82·77-s + 0.900·79-s + 1/9·81-s − 0.402·99-s − 1/5·100-s + 1.54·107-s + 0.766·109-s + 0.377·112-s − 0.376·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(213444\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(13.6093\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 213444,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.293336294\)
\(L(\frac12)\) \(\approx\) \(2.293336294\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
11$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
13$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.13.a_as
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.17.a_ao
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.29.aq_eo
31$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.31.a_ack
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.37.m_eg
41$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.41.a_abe
43$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.43.q_fe
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.53.au_hi
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.59.a_cs
61$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.61.a_bu
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.71.ay_kk
73$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.73.a_s
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.79.ai_gs
83$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.83.a_k
89$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.89.a_aby
97$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.97.a_eg
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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

−8.665096863199730193528536428999, −8.524253019447863055235997230797, −8.188769389892575152529270937218, −7.70217273809722583435498987335, −7.12672404077327727776261554757, −6.74185961184153339193775662900, −6.23969165149518831172612641577, −5.32403103939328504554919900736, −5.09911818009051704782014362269, −4.78243254016688788309840525166, −3.92529785180152174014596626593, −3.28705613589508262967439884468, −2.41205735758110562376113309934, −1.99746960372604892155109398017, −1.00441622224102862653037289833, 1.00441622224102862653037289833, 1.99746960372604892155109398017, 2.41205735758110562376113309934, 3.28705613589508262967439884468, 3.92529785180152174014596626593, 4.78243254016688788309840525166, 5.09911818009051704782014362269, 5.32403103939328504554919900736, 6.23969165149518831172612641577, 6.74185961184153339193775662900, 7.12672404077327727776261554757, 7.70217273809722583435498987335, 8.188769389892575152529270937218, 8.524253019447863055235997230797, 8.665096863199730193528536428999

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