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Properties

Label 4-1707552-1.1-c1e2-0-15
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 − 4·7-s − 8-s + 9-s − 4·11-s + 4·14-s + 16-s − 18-s + 4·22-s − 10·23-s − 8·25-s − 4·28-s − 32-s + 36-s + 4·37-s − 2·43-s − 4·44-s + 10·46-s + 9·49-s + 8·50-s + 26·53-s + 4·56-s − 4·63-s + 64-s + 12·67-s − 2·71-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 1.20·11-s + 1.06·14-s + 1/4·16-s − 0.235·18-s + 0.852·22-s − 2.08·23-s − 8/5·25-s − 0.755·28-s − 0.176·32-s + 1/6·36-s + 0.657·37-s − 0.304·43-s − 0.603·44-s + 1.47·46-s + 9/7·49-s + 1.13·50-s + 3.57·53-s + 0.534·56-s − 0.503·63-s + 1/8·64-s + 1.46·67-s − 0.237·71-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$C_2$ \( 1 + 4 T + p T^{2} \)
11$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.5.a_i
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.17.a_e
19$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.19.a_q
23$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.k_ck
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.a_bq
31$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.31.a_q
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ae_ck
41$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.41.a_au
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.43.c_ck
47$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \) 2.47.a_adc
53$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) 2.53.aba_ko
59$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.59.a_k
61$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.61.a_bm
67$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.67.am_fy
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.71.c_fm
73$C_2^2$ \( 1 - 104 T^{2} + p^{2} T^{4} \) 2.73.a_aea
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.ae_ew
83$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \) 2.83.a_ce
89$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \) 2.89.a_co
97$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.97.a_ag
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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

−7.57851729497927355701736584210, −7.38184401092477047311562994700, −6.76927208012184346348983871903, −6.37536178160567248909529297497, −6.02249367621967807367301672794, −5.46225219791173253551851583448, −5.34380642146194802099365731188, −4.20930433067046279471336693936, −4.05567747782485019571367873580, −3.49576308008448780160317748545, −2.87364880000081578812764085938, −2.25863256077370037761867912205, −1.99800060103855743970211139741, −0.73798703027665168856944263212, 0, 0.73798703027665168856944263212, 1.99800060103855743970211139741, 2.25863256077370037761867912205, 2.87364880000081578812764085938, 3.49576308008448780160317748545, 4.05567747782485019571367873580, 4.20930433067046279471336693936, 5.34380642146194802099365731188, 5.46225219791173253551851583448, 6.02249367621967807367301672794, 6.37536178160567248909529297497, 6.76927208012184346348983871903, 7.38184401092477047311562994700, 7.57851729497927355701736584210

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