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Properties

Label 4-462e2-1.1-c1e2-0-5
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 + 2·7-s − 9-s + 2·11-s + 16-s + 12·23-s − 2·25-s + 2·28-s − 10·29-s − 36-s + 2·37-s − 4·43-s + 2·44-s − 3·49-s + 10·53-s − 2·63-s + 64-s + 2·67-s − 2·71-s + 4·77-s + 32·79-s + 81-s + 12·92-s − 2·99-s − 2·100-s + 2·107-s + 32·109-s + ⋯
L(s)  = 1  + 1/2·4-s + 0.755·7-s − 1/3·9-s + 0.603·11-s + 1/4·16-s + 2.50·23-s − 2/5·25-s + 0.377·28-s − 1.85·29-s − 1/6·36-s + 0.328·37-s − 0.609·43-s + 0.301·44-s − 3/7·49-s + 1.37·53-s − 0.251·63-s + 1/8·64-s + 0.244·67-s − 0.237·71-s + 0.455·77-s + 3.60·79-s + 1/9·81-s + 1.25·92-s − 0.201·99-s − 1/5·100-s + 0.193·107-s + 3.06·109-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

\(L(1)\) \(\approx\) \(2.229530679\)
\(L(\frac12)\) \(\approx\) \(2.229530679\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
11$C_2$ \( 1 - 2 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.17.a_u
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.23.am_da
29$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.k_cw
31$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.31.a_ae
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.ac_ba
41$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.41.a_ae
43$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.e_di
47$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \) 2.47.a_bs
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.ak_de
59$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.59.a_acg
61$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.61.a_ac
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.67.ac_ew
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.71.c_fm
73$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.73.a_aq
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.79.abg_py
83$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.83.a_aec
89$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.89.a_be
97$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.97.a_da
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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

−9.071847077468094801685981044147, −8.573608111482918720490368649868, −8.144145438448620747718683711539, −7.45310891683710979012343198366, −7.26645047823184585184484401405, −6.70827284064146508683978523642, −6.13370449196136291440850350111, −5.63447221482099229690466066259, −4.98354907985743638927001979494, −4.75269561134398220102237601082, −3.70425039823523205749891936368, −3.45801353566867138481956888073, −2.52758932872561215881814473405, −1.88606036963751703925724745625, −1.00124895632195069334182774366, 1.00124895632195069334182774366, 1.88606036963751703925724745625, 2.52758932872561215881814473405, 3.45801353566867138481956888073, 3.70425039823523205749891936368, 4.75269561134398220102237601082, 4.98354907985743638927001979494, 5.63447221482099229690466066259, 6.13370449196136291440850350111, 6.70827284064146508683978523642, 7.26645047823184585184484401405, 7.45310891683710979012343198366, 8.144145438448620747718683711539, 8.573608111482918720490368649868, 9.071847077468094801685981044147

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