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Properties

Label 4-3024e2-1.1-c1e2-0-19
Degree $4$
Conductor $9144576$
Sign $1$
Analytic cond. $583.066$
Root an. cond. $4.91393$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s + 7-s + 6·11-s − 2·13-s − 12·17-s + 14·19-s − 3·23-s + 5·25-s + 6·29-s + 2·31-s − 3·35-s + 4·37-s + 2·43-s − 12·53-s − 18·55-s − 5·61-s + 6·65-s + 8·67-s + 6·71-s + 4·73-s + 6·77-s + 5·79-s − 12·83-s + 36·85-s − 2·91-s − 42·95-s − 2·97-s + ⋯
L(s)  = 1  − 1.34·5-s + 0.377·7-s + 1.80·11-s − 0.554·13-s − 2.91·17-s + 3.21·19-s − 0.625·23-s + 25-s + 1.11·29-s + 0.359·31-s − 0.507·35-s + 0.657·37-s + 0.304·43-s − 1.64·53-s − 2.42·55-s − 0.640·61-s + 0.744·65-s + 0.977·67-s + 0.712·71-s + 0.468·73-s + 0.683·77-s + 0.562·79-s − 1.31·83-s + 3.90·85-s − 0.209·91-s − 4.30·95-s − 0.203·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9144576\)    =    \(2^{8} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(583.066\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9144576,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.690079146\)
\(L(\frac12)\) \(\approx\) \(1.690079146\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
7$C_2$ \( 1 - T + T^{2} \)
good5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_e
11$C_2^2$ \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_z
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.c_aj
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.17.m_cs
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.19.ao_dj
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.23.d_ao
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_h
31$C_2^2$ \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.31.ac_abb
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.41.a_abp
43$C_2^2$ \( 1 - 2 T - 39 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.43.ac_abn
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2^2$ \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.61.f_abk
67$C_2^2$ \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_ad
71$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.71.ag_fv
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2^2$ \( 1 - 5 T - 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.79.af_acc
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_cj
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.97.c_adp
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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

−9.383126808480410077594299163292, −8.374203506294934692410051619554, −8.157009355248475792742284741318, −7.900083646337864072267200587299, −7.34642532321144877470488027067, −6.93494401502035205732516124128, −6.89384466156643867158957973411, −6.31459767718481154988040146181, −6.07175033094876052436970452198, −5.22078132431895077307511529352, −5.04125930076326715285816824818, −4.42231503339086593042553156349, −4.35764082976148223710878118128, −3.77975240139713523891249568697, −3.50402980389435609133345168730, −2.75735760847672058934669876033, −2.55299713738741700113025647450, −1.60510241191156399030294964624, −1.20092829442801032971968923783, −0.45477622755877449628922801115, 0.45477622755877449628922801115, 1.20092829442801032971968923783, 1.60510241191156399030294964624, 2.55299713738741700113025647450, 2.75735760847672058934669876033, 3.50402980389435609133345168730, 3.77975240139713523891249568697, 4.35764082976148223710878118128, 4.42231503339086593042553156349, 5.04125930076326715285816824818, 5.22078132431895077307511529352, 6.07175033094876052436970452198, 6.31459767718481154988040146181, 6.89384466156643867158957973411, 6.93494401502035205732516124128, 7.34642532321144877470488027067, 7.900083646337864072267200587299, 8.157009355248475792742284741318, 8.374203506294934692410051619554, 9.383126808480410077594299163292

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