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Properties

Label 4-1050e2-1.1-c1e2-0-28
Degree $4$
Conductor $1102500$
Sign $1$
Analytic cond. $70.2963$
Root an. cond. $2.89556$
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  + 2-s − 3-s − 6-s + 4·7-s − 8-s + 5·11-s + 10·13-s + 4·14-s − 16-s − 4·17-s + 7·19-s − 4·21-s + 5·22-s + 23-s + 24-s + 10·26-s + 27-s + 2·31-s − 5·33-s − 4·34-s + 37-s + 7·38-s − 10·39-s + 10·41-s − 4·42-s − 24·43-s + 46-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 0.408·6-s + 1.51·7-s − 0.353·8-s + 1.50·11-s + 2.77·13-s + 1.06·14-s − 1/4·16-s − 0.970·17-s + 1.60·19-s − 0.872·21-s + 1.06·22-s + 0.208·23-s + 0.204·24-s + 1.96·26-s + 0.192·27-s + 0.359·31-s − 0.870·33-s − 0.685·34-s + 0.164·37-s + 1.13·38-s − 1.60·39-s + 1.56·41-s − 0.617·42-s − 3.65·43-s + 0.147·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1102500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(70.2963\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1102500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.923958618\)
\(L(\frac12)\) \(\approx\) \(3.923958618\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 + T + T^{2} \)
5 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good11$C_2^2$ \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.11.af_o
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.13.ak_bz
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.17.e_ab
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.ah_be
23$C_2^2$ \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) 2.23.ab_aw
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2^2$ \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.31.ac_abb
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.ab_abk
41$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.41.ak_ed
43$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.43.y_iw
47$C_2^2$ \( 1 + 11 T + 74 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.47.l_cw
53$C_2^2$ \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.53.j_bc
59$C_2^2$ \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_abr
61$C_2^2$ \( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_abt
67$C_2^2$ \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.67.m_cz
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.71.ae_fq
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.73.ak_bb
79$C_2^2$ \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.79.am_cn
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.83.ay_ly
89$C_2^2$ \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.89.o_ed
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.97.aq_jy
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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

−10.17684586192007170472353280803, −9.714730648138680166752435378692, −9.013140524327360030490668952504, −9.004742155934751503145198306489, −8.522554300842937996223940444423, −7.945666662056008116658145563784, −7.84631061255104328484130762315, −7.00176297916936316813590296574, −6.41198847974206336309067270989, −6.31862866342823298207153928087, −6.00367422045603836064956133651, −5.14352390995270661629306157375, −5.00057552894100604437023974583, −4.53683789271397133296162415601, −3.96873437673761699233149625100, −3.34044381846113541302232537787, −3.32940374808960685605755158848, −1.94506973807200219791793877030, −1.44939202141652908605457048114, −0.982088696251283538707303216707, 0.982088696251283538707303216707, 1.44939202141652908605457048114, 1.94506973807200219791793877030, 3.32940374808960685605755158848, 3.34044381846113541302232537787, 3.96873437673761699233149625100, 4.53683789271397133296162415601, 5.00057552894100604437023974583, 5.14352390995270661629306157375, 6.00367422045603836064956133651, 6.31862866342823298207153928087, 6.41198847974206336309067270989, 7.00176297916936316813590296574, 7.84631061255104328484130762315, 7.945666662056008116658145563784, 8.522554300842937996223940444423, 9.004742155934751503145198306489, 9.013140524327360030490668952504, 9.714730648138680166752435378692, 10.17684586192007170472353280803

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