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Properties

Label 4-1323e2-1.1-c1e2-0-12
Degree $4$
Conductor $1750329$
Sign $1$
Analytic cond. $111.602$
Root an. cond. $3.25026$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 4·4-s + 3·5-s − 3·8-s − 9·10-s + 3·11-s + 3·13-s + 3·16-s − 6·17-s + 12·20-s − 9·22-s + 9·23-s + 5·25-s − 9·26-s + 9·29-s − 6·31-s − 6·32-s + 18·34-s + 14·37-s − 9·40-s + 3·41-s − 43-s + 12·44-s − 27·46-s − 15·50-s + 12·52-s + 9·55-s + ⋯
L(s)  = 1  − 2.12·2-s + 2·4-s + 1.34·5-s − 1.06·8-s − 2.84·10-s + 0.904·11-s + 0.832·13-s + 3/4·16-s − 1.45·17-s + 2.68·20-s − 1.91·22-s + 1.87·23-s + 25-s − 1.76·26-s + 1.67·29-s − 1.07·31-s − 1.06·32-s + 3.08·34-s + 2.30·37-s − 1.42·40-s + 0.468·41-s − 0.152·43-s + 1.80·44-s − 3.98·46-s − 2.12·50-s + 1.66·52-s + 1.21·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1750329\)    =    \(3^{6} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(111.602\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1750329,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.106388903\)
\(L(\frac12)\) \(\approx\) \(1.106388903\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3 \( 1 \)
7 \( 1 \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.2.d_f
5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.5.ad_e
11$C_2^2$ \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_o
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.ad_q
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.17.g_br
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.a_al
23$C_2^2$ \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.23.aj_by
29$C_2^2$ \( 1 - 9 T + 56 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.29.aj_ce
31$C_2^2$ \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.31.g_br
37$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.37.ao_et
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.b_abq
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \) 2.53.a_abf
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2^2$ \( 1 - 24 T + 253 T^{2} - 24 p T^{3} + p^{2} T^{4} \) 2.61.ay_jt
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_abz
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.71.a_afa
73$C_2^2$ \( 1 - 119 T^{2} + p^{2} T^{4} \) 2.73.a_aep
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_ap
83$C_2^2$ \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.83.p_fm
89$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.89.g_hf
97$C_2^2$ \( 1 - 3 T + 100 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.97.ad_dw
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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

−9.697778134602527532931990691281, −9.452343792465186527144318567497, −9.027047913070435534133398151140, −8.715630154397928085371836593257, −8.359132464722967223231435292206, −8.306552609302574962413431944206, −7.31308050027186436270258386943, −7.13965319030429974490050212771, −6.64056105573029073589598917437, −6.37804935225052383484084021853, −5.71219349024419316037135419274, −5.60072337566334986200317027605, −4.59573602895056667858747484345, −4.47294253263176121643558856221, −3.60755652207958569346407357934, −2.94488852275526794249830348716, −2.39512660561262032965554588682, −1.81921505335063145846772891630, −1.07148710939418932926165939560, −0.798811610972492510804057422554, 0.798811610972492510804057422554, 1.07148710939418932926165939560, 1.81921505335063145846772891630, 2.39512660561262032965554588682, 2.94488852275526794249830348716, 3.60755652207958569346407357934, 4.47294253263176121643558856221, 4.59573602895056667858747484345, 5.60072337566334986200317027605, 5.71219349024419316037135419274, 6.37804935225052383484084021853, 6.64056105573029073589598917437, 7.13965319030429974490050212771, 7.31308050027186436270258386943, 8.306552609302574962413431944206, 8.359132464722967223231435292206, 8.715630154397928085371836593257, 9.027047913070435534133398151140, 9.452343792465186527144318567497, 9.697778134602527532931990691281

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