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Properties

Label 4-1707552-1.1-c1e2-0-43
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-s + 4-s + 2·7-s − 8-s + 9-s + 2·11-s − 2·14-s + 16-s − 18-s − 2·22-s − 4·23-s − 2·25-s + 2·28-s + 6·29-s − 32-s + 36-s − 2·37-s + 4·43-s + 2·44-s + 4·46-s − 3·49-s + 2·50-s − 10·53-s − 2·56-s − 6·58-s + 2·63-s + 64-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.534·14-s + 1/4·16-s − 0.235·18-s − 0.426·22-s − 0.834·23-s − 2/5·25-s + 0.377·28-s + 1.11·29-s − 0.176·32-s + 1/6·36-s − 0.328·37-s + 0.609·43-s + 0.301·44-s + 0.589·46-s − 3/7·49-s + 0.282·50-s − 1.37·53-s − 0.267·56-s − 0.787·58-s + 0.251·63-s + 1/8·64-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$C_1$ \( 1 + T \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 - 2 T + p T^{2} \)
11$C_2$ \( 1 - 2 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.17.a_au
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.e_o
29$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.ag_bq
31$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.31.a_e
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.c_ba
41$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.41.a_e
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.43.ae_di
47$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \) 2.47.a_abs
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.k_de
59$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.59.a_cg
61$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.61.a_c
67$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.g_fm
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.71.c_fm
73$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.73.a_q
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.79.bg_py
83$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.83.a_aec
89$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.89.a_be
97$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.97.a_da
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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

−7.63520934784353764045914817195, −7.28844652514577506426838994433, −6.88662858411620937049278017415, −6.39901022084612047489984833222, −5.89331344552768592463477664762, −5.70065133313962609323012091723, −4.76848266992381879176006853234, −4.65602487391672324701510739492, −4.04876198895936488180353403207, −3.50295953105173701545459249679, −2.87437841864846219496524500527, −2.28159645018256982561324225080, −1.56588667258821092141210086851, −1.22146856538236981222675445965, 0, 1.22146856538236981222675445965, 1.56588667258821092141210086851, 2.28159645018256982561324225080, 2.87437841864846219496524500527, 3.50295953105173701545459249679, 4.04876198895936488180353403207, 4.65602487391672324701510739492, 4.76848266992381879176006853234, 5.70065133313962609323012091723, 5.89331344552768592463477664762, 6.39901022084612047489984833222, 6.88662858411620937049278017415, 7.28844652514577506426838994433, 7.63520934784353764045914817195

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