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Properties

Label 4-93312-1.1-c1e2-0-2
Degree $4$
Conductor $93312$
Sign $1$
Analytic cond. $5.94965$
Root an. cond. $1.56179$
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  − 2-s + 4-s − 2·7-s − 8-s + 2·14-s + 16-s − 3·23-s + 8·25-s − 2·28-s + 10·31-s − 32-s + 6·41-s + 3·46-s − 6·47-s − 2·49-s − 8·50-s + 2·56-s − 10·62-s + 64-s − 18·71-s + 4·73-s + 7·79-s − 6·82-s + 9·89-s − 3·92-s + 6·94-s − 11·97-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s + 0.534·14-s + 1/4·16-s − 0.625·23-s + 8/5·25-s − 0.377·28-s + 1.79·31-s − 0.176·32-s + 0.937·41-s + 0.442·46-s − 0.875·47-s − 2/7·49-s − 1.13·50-s + 0.267·56-s − 1.27·62-s + 1/8·64-s − 2.13·71-s + 0.468·73-s + 0.787·79-s − 0.662·82-s + 0.953·89-s − 0.312·92-s + 0.618·94-s − 1.11·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(93312\)    =    \(2^{7} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(5.94965\)
Root analytic conductor: \(1.56179\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 93312,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9389865857\)
\(L(\frac12)\) \(\approx\) \(0.9389865857\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
3 \( 1 \)
good5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.5.a_ai
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.c_g
11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.11.a_e
13$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.13.a_ai
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.d_bu
29$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \) 2.29.a_abg
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.31.ak_dj
37$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.37.a_q
41$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.41.ag_cd
43$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.43.a_bo
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.g_dq
53$C_2^2$ \( 1 + 88 T^{2} + p^{2} T^{4} \) 2.53.a_dk
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.a_de
61$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.61.a_aby
67$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \) 2.67.a_cm
71$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.71.s_hf
73$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.73.ae_ek
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.79.ah_fu
83$C_2^2$ \( 1 + 148 T^{2} + p^{2} T^{4} \) 2.83.a_fs
89$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.aj_dk
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.97.l_gm
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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

−9.603384002109920925484166971787, −9.183673376102949056621935040645, −8.610028361000783704310897032066, −8.279133309389622600251427438967, −7.65274014464953476243705691750, −7.15783105795791171099208071963, −6.55789704255062500582585448580, −6.24150487485003178520648701843, −5.66650334980828507799304208986, −4.78611486436286309893354863873, −4.36075045479923626167253518031, −3.30513816527997452318395215051, −2.94741495006642520553938109985, −2.02243932175308410887290963272, −0.818618891798656203725184699877, 0.818618891798656203725184699877, 2.02243932175308410887290963272, 2.94741495006642520553938109985, 3.30513816527997452318395215051, 4.36075045479923626167253518031, 4.78611486436286309893354863873, 5.66650334980828507799304208986, 6.24150487485003178520648701843, 6.55789704255062500582585448580, 7.15783105795791171099208071963, 7.65274014464953476243705691750, 8.279133309389622600251427438967, 8.610028361000783704310897032066, 9.183673376102949056621935040645, 9.603384002109920925484166971787

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