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Properties

Label 4-93312-1.1-c1e2-0-34
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5·7-s + 8-s − 5·14-s + 16-s − 15·23-s − 7·25-s − 5·28-s − 2·31-s + 32-s + 12·41-s − 15·46-s − 3·47-s + 7·49-s − 7·50-s − 5·56-s − 2·62-s + 64-s + 9·71-s − 8·73-s − 11·79-s + 12·82-s + 9·89-s − 15·92-s − 3·94-s − 14·97-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.88·7-s + 0.353·8-s − 1.33·14-s + 1/4·16-s − 3.12·23-s − 7/5·25-s − 0.944·28-s − 0.359·31-s + 0.176·32-s + 1.87·41-s − 2.21·46-s − 0.437·47-s + 49-s − 0.989·50-s − 0.668·56-s − 0.254·62-s + 1/8·64-s + 1.06·71-s − 0.936·73-s − 1.23·79-s + 1.32·82-s + 0.953·89-s − 1.56·92-s − 0.309·94-s − 1.42·97-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: \(1\)
Selberg data: \((4,\ 93312,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
3 \( 1 \)
good5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.5.a_h
7$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.f_s
11$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.11.a_ai
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.13.a_b
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.a_z
19$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.19.a_au
23$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.23.p_dw
29$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.29.a_af
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.31.c_as
37$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.37.a_bi
41$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.41.am_de
43$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \) 2.43.a_acq
47$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.47.d_bo
53$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.53.a_af
59$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.59.a_bc
61$C_2^2$ \( 1 + 85 T^{2} + p^{2} T^{4} \) 2.61.a_dh
67$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \) 2.67.a_cm
71$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.71.aj_ge
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.i_ew
79$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.l_gm
83$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \) 2.83.a_adu
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$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.o_ja
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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

−9.652714422744095323466797360572, −9.027659096749185878742715215599, −8.225571133318294688989128787892, −7.81703477466188316464245955479, −7.33220847183364183467575379900, −6.61188979234268611173870798333, −6.13353325696699756481986466494, −5.96267241280273743869494244056, −5.36003727661739443711228678585, −4.25409564691026793235370079339, −3.99502876875398050428326696477, −3.40242420560698287174088785935, −2.63835594809174172822601552921, −1.90599422533204285651276085720, 0, 1.90599422533204285651276085720, 2.63835594809174172822601552921, 3.40242420560698287174088785935, 3.99502876875398050428326696477, 4.25409564691026793235370079339, 5.36003727661739443711228678585, 5.96267241280273743869494244056, 6.13353325696699756481986466494, 6.61188979234268611173870798333, 7.33220847183364183467575379900, 7.81703477466188316464245955479, 8.225571133318294688989128787892, 9.027659096749185878742715215599, 9.652714422744095323466797360572

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