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Properties

Label 4-42e4-1.1-c1e2-0-15
Degree $4$
Conductor $3111696$
Sign $1$
Analytic cond. $198.404$
Root an. cond. $3.75308$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 3·5-s + 6·9-s − 3·11-s − 13-s + 9·15-s + 6·17-s − 4·19-s + 3·23-s + 5·25-s + 9·27-s − 3·29-s − 10·31-s − 9·33-s − 2·37-s − 3·39-s + 3·41-s + 43-s + 18·45-s + 18·47-s + 18·51-s + 6·53-s − 9·55-s − 12·57-s + 6·59-s + 26·61-s − 3·65-s + ⋯
L(s)  = 1  + 1.73·3-s + 1.34·5-s + 2·9-s − 0.904·11-s − 0.277·13-s + 2.32·15-s + 1.45·17-s − 0.917·19-s + 0.625·23-s + 25-s + 1.73·27-s − 0.557·29-s − 1.79·31-s − 1.56·33-s − 0.328·37-s − 0.480·39-s + 0.468·41-s + 0.152·43-s + 2.68·45-s + 2.62·47-s + 2.52·51-s + 0.824·53-s − 1.21·55-s − 1.58·57-s + 0.781·59-s + 3.32·61-s − 0.372·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3111696\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(198.404\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3111696,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.395874455\)
\(L(\frac12)\) \(\approx\) \(6.395874455\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.5.ad_e
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_ac
13$C_2^2$ \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) 2.13.b_am
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_t
19$C_2^2$ \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_ad
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.23.ad_ao
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.29.d_au
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.31.k_dj
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.37.c_abh
41$C_2^2$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.41.ad_abg
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) 2.43.ab_abq
47$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.47.as_gt
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.53.ag_ar
59$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.59.ag_ex
61$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \) 2.61.aba_lf
67$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.67.o_hb
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.73.k_bb
79$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.79.aw_kt
83$C_2^2$ \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.83.j_ac
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.89.ag_acb
97$C_2^2$ \( 1 - 11 T + 24 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.97.al_y
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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

−9.444803248046898725972062443729, −9.006892466279282178175879640765, −8.717039056589033700261647410812, −8.612481923755314808691004583157, −7.78879172725979189256679975026, −7.62176417970067082748136768778, −7.24555731338226662547463188010, −6.96630415067636708266318504514, −6.28727758504548404321621562170, −5.66175014176923779218161155225, −5.51517353989944088224359349341, −5.19654731104344314686480371203, −4.23435876004542937315464188037, −4.16456234783465249932817469006, −3.34135657814592957674806550639, −3.10482898655835285615667472816, −2.34443467113015809225062503324, −2.25046092660961627731653106877, −1.66932601825441563141567650067, −0.851548885483323495055053457413, 0.851548885483323495055053457413, 1.66932601825441563141567650067, 2.25046092660961627731653106877, 2.34443467113015809225062503324, 3.10482898655835285615667472816, 3.34135657814592957674806550639, 4.16456234783465249932817469006, 4.23435876004542937315464188037, 5.19654731104344314686480371203, 5.51517353989944088224359349341, 5.66175014176923779218161155225, 6.28727758504548404321621562170, 6.96630415067636708266318504514, 7.24555731338226662547463188010, 7.62176417970067082748136768778, 7.78879172725979189256679975026, 8.612481923755314808691004583157, 8.717039056589033700261647410812, 9.006892466279282178175879640765, 9.444803248046898725972062443729

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