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Properties

Label 4-125312-1.1-c1e2-0-3
Degree $4$
Conductor $125312$
Sign $-1$
Analytic cond. $7.99000$
Root an. cond. $1.68126$
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 + 8-s − 5·9-s + 16-s − 9·17-s − 5·18-s − 3·19-s − 4·25-s + 32-s − 9·34-s − 5·36-s − 3·38-s + 21·41-s − 8·43-s − 2·49-s − 4·50-s + 64-s − 30·67-s − 9·68-s − 5·72-s − 10·73-s − 3·76-s + 16·81-s + 21·82-s − 4·83-s − 8·86-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 5/3·9-s + 1/4·16-s − 2.18·17-s − 1.17·18-s − 0.688·19-s − 4/5·25-s + 0.176·32-s − 1.54·34-s − 5/6·36-s − 0.486·38-s + 3.27·41-s − 1.21·43-s − 2/7·49-s − 0.565·50-s + 1/8·64-s − 3.66·67-s − 1.09·68-s − 0.589·72-s − 1.17·73-s − 0.344·76-s + 16/9·81-s + 2.31·82-s − 0.439·83-s − 0.862·86-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(125312\)    =    \(2^{7} \cdot 11 \cdot 89\)
Sign: $-1$
Analytic conductor: \(7.99000\)
Root analytic conductor: \(1.68126\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 125312,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
11$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + T + p T^{2} ) \)
89$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 5 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.5.a_e
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
13$C_2^2$ \( 1 + 21 T^{2} + p^{2} T^{4} \) 2.13.a_v
17$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.17.j_bw
19$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.d_bi
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.a_as
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.31.a_aby
37$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.37.a_an
41$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) 2.41.av_hi
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.47.a_aw
53$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.53.a_abi
59$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.59.a_ek
61$C_2^2$ \( 1 + 77 T^{2} + p^{2} T^{4} \) 2.61.a_cz
67$C_2$$\times$$C_2$ \( ( 1 + 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.be_nu
71$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.71.a_abj
73$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.k_go
79$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \) 2.79.a_co
83$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.e_cs
97$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.ac_he
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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

−9.091730872925010345727983119432, −8.702968314483042109388777318203, −8.264516971564103824977232912425, −7.62861958722954994325091533549, −7.17722910636867387305044864844, −6.37335327577639281284003500408, −6.14319039511301386322193595320, −5.72695460565078276709379865439, −5.00520370037502466424334334994, −4.33016372178985003048437040014, −4.06657384710885970248970434080, −2.97344788145819386256167474629, −2.63719961338146612310474491788, −1.85605136848062850931543318606, 0, 1.85605136848062850931543318606, 2.63719961338146612310474491788, 2.97344788145819386256167474629, 4.06657384710885970248970434080, 4.33016372178985003048437040014, 5.00520370037502466424334334994, 5.72695460565078276709379865439, 6.14319039511301386322193595320, 6.37335327577639281284003500408, 7.17722910636867387305044864844, 7.62861958722954994325091533549, 8.264516971564103824977232912425, 8.702968314483042109388777318203, 9.091730872925010345727983119432

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