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Properties

Label 4-1512e2-1.1-c1e2-0-25
Degree $4$
Conductor $2286144$
Sign $1$
Analytic cond. $145.766$
Root an. cond. $3.47467$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 8·11-s − 4·16-s + 6·17-s − 16·19-s + 16·22-s − 9·25-s + 8·32-s − 12·34-s + 32·38-s + 2·41-s + 22·43-s − 16·44-s + 49-s + 18·50-s − 30·59-s − 8·64-s − 16·67-s + 12·68-s + 12·73-s − 32·76-s − 4·82-s − 18·83-s − 44·86-s + 4·89-s + 24·97-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 2.41·11-s − 16-s + 1.45·17-s − 3.67·19-s + 3.41·22-s − 9/5·25-s + 1.41·32-s − 2.05·34-s + 5.19·38-s + 0.312·41-s + 3.35·43-s − 2.41·44-s + 1/7·49-s + 2.54·50-s − 3.90·59-s − 64-s − 1.95·67-s + 1.45·68-s + 1.40·73-s − 3.67·76-s − 0.441·82-s − 1.97·83-s − 4.74·86-s + 0.423·89-s + 2.43·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2286144\)    =    \(2^{6} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(145.766\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2286144,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + p T + p T^{2} \)
3 \( 1 \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.5.a_j
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.11.i_bm
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.19.q_dy
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.a_bq
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.31.a_ba
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.37.a_cn
41$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.41.ac_df
43$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.43.aw_hz
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.47.a_n
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \) 2.59.be_nf
61$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.61.a_ec
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.67.q_hq
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.79.a_gb
83$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.83.s_jn
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.89.ae_ha
97$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.97.ay_na
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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

−7.43939934466159987675241180651, −7.30481690380781283806945348292, −6.27218716655140034782382797882, −5.97786895192084747528753482689, −5.97692172450357958528696118657, −5.09187816380224762265303345894, −4.60343557040003079873210765536, −4.28481394661801806092561213309, −3.69161989929682443753436153368, −2.83738057526983250492336310985, −2.34130595001443635884001685756, −2.13554297795060365862589896807, −1.26608620880996381438829664536, 0, 0, 1.26608620880996381438829664536, 2.13554297795060365862589896807, 2.34130595001443635884001685756, 2.83738057526983250492336310985, 3.69161989929682443753436153368, 4.28481394661801806092561213309, 4.60343557040003079873210765536, 5.09187816380224762265303345894, 5.97692172450357958528696118657, 5.97786895192084747528753482689, 6.27218716655140034782382797882, 7.30481690380781283806945348292, 7.43939934466159987675241180651

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