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Properties

Label 4-1707552-1.1-c1e2-0-42
Degree $4$
Conductor $1707552$
Sign $-1$
Analytic cond. $108.874$
Root an. cond. $3.23021$
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  − 2·4-s + 7-s + 9-s + 4·11-s + 4·16-s − 2·23-s + 7·25-s − 2·28-s − 6·29-s − 2·36-s − 8·37-s − 6·43-s − 8·44-s − 6·49-s + 2·53-s + 63-s − 8·64-s + 67-s − 14·71-s + 4·77-s + 22·79-s + 81-s + 4·92-s + 4·99-s − 14·100-s − 14·107-s − 5·109-s + ⋯
L(s)  = 1  − 4-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 16-s − 0.417·23-s + 7/5·25-s − 0.377·28-s − 1.11·29-s − 1/3·36-s − 1.31·37-s − 0.914·43-s − 1.20·44-s − 6/7·49-s + 0.274·53-s + 0.125·63-s − 64-s + 0.122·67-s − 1.66·71-s + 0.455·77-s + 2.47·79-s + 1/9·81-s + 0.417·92-s + 0.402·99-s − 7/5·100-s − 1.35·107-s − 0.478·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1707552 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1707552 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1707552\)    =    \(2^{5} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(108.874\)
Root analytic conductor: \(3.23021\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1707552,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + p T^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 - T + p T^{2} \)
11$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \) 2.5.a_ah
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2^2$ \( 1 + 29 T^{2} + p^{2} T^{4} \) 2.17.a_bd
19$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.19.a_az
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.c_w
29$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.g_co
31$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \) 2.31.a_p
37$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.37.i_bp
41$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.41.a_aw
43$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.43.g_db
47$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.47.a_ar
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.ac_de
59$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.59.a_bl
61$C_2^2$ \( 1 + 9 T^{2} + p^{2} T^{4} \) 2.61.a_j
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.67.ab_es
71$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.o_hi
73$C_2^2$ \( 1 + 115 T^{2} + p^{2} T^{4} \) 2.73.a_el
79$C_2$$\times$$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.79.aw_jj
83$C_2^2$ \( 1 + 79 T^{2} + p^{2} T^{4} \) 2.83.a_db
89$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.89.a_l
97$C_2^2$ \( 1 + 117 T^{2} + p^{2} T^{4} \) 2.97.a_en
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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

−7.61893543170843916071894921244, −7.26614969297607450879694554648, −6.67462895486749139285207411600, −6.46751358104590098821941953451, −5.84311522766738279123103516873, −5.31704574996115538717927190226, −4.96305678192425083863033529776, −4.56127271447930976027443374773, −4.01829869616617289764214597371, −3.58781784922424286180081538943, −3.25309713598141003968806378835, −2.33900855794272030706500907035, −1.56555337111917917913802024638, −1.14757780875046504716833459048, 0, 1.14757780875046504716833459048, 1.56555337111917917913802024638, 2.33900855794272030706500907035, 3.25309713598141003968806378835, 3.58781784922424286180081538943, 4.01829869616617289764214597371, 4.56127271447930976027443374773, 4.96305678192425083863033529776, 5.31704574996115538717927190226, 5.84311522766738279123103516873, 6.46751358104590098821941953451, 6.67462895486749139285207411600, 7.26614969297607450879694554648, 7.61893543170843916071894921244

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