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Properties

Label 4-1323e2-1.1-c1e2-0-2
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  − 2·2-s − 4-s + 8·8-s − 4·11-s − 7·16-s + 8·22-s − 12·23-s + 6·25-s − 4·29-s − 14·32-s + 6·37-s − 2·43-s + 4·44-s + 24·46-s − 12·50-s + 12·53-s + 8·58-s + 35·64-s + 14·67-s − 12·74-s + 22·79-s + 4·86-s − 32·88-s + 12·92-s − 6·100-s − 24·106-s − 4·107-s + ⋯
L(s)  = 1  − 1.41·2-s − 1/2·4-s + 2.82·8-s − 1.20·11-s − 7/4·16-s + 1.70·22-s − 2.50·23-s + 6/5·25-s − 0.742·29-s − 2.47·32-s + 0.986·37-s − 0.304·43-s + 0.603·44-s + 3.53·46-s − 1.69·50-s + 1.64·53-s + 1.05·58-s + 35/8·64-s + 1.71·67-s − 1.39·74-s + 2.47·79-s + 0.431·86-s − 3.41·88-s + 1.25·92-s − 3/5·100-s − 2.33·106-s − 0.386·107-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

\(L(1)\) \(\approx\) \(0.3837233950\)
\(L(\frac12)\) \(\approx\) \(0.3837233950\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.2.c_f
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.11.e_ba
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.13.a_z
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.23.m_de
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.31.a_cb
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.37.ag_df
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.a_da
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.43.c_dj
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.a_de
61$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.61.a_dt
67$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.67.ao_hb
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.a_eg
79$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.79.aw_kt
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.89.a_gg
97$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.97.a_ej
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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

−8.116653920015147863681030485165, −7.47906214811694943254418977697, −7.42647965860644167776315724949, −6.69301809340706125234074868915, −6.14623737900759535985097643782, −5.55497953150978441638676387304, −5.22174626816378645292658033645, −4.77492261402819605358514038878, −4.26459885949450872653953319987, −3.85297585812455508834915893838, −3.34295702159287177762130169073, −2.27467032412786158333398018626, −2.07336320253444604618926237883, −1.06074849095375840135977622104, −0.40116959550324554090187768272, 0.40116959550324554090187768272, 1.06074849095375840135977622104, 2.07336320253444604618926237883, 2.27467032412786158333398018626, 3.34295702159287177762130169073, 3.85297585812455508834915893838, 4.26459885949450872653953319987, 4.77492261402819605358514038878, 5.22174626816378645292658033645, 5.55497953150978441638676387304, 6.14623737900759535985097643782, 6.69301809340706125234074868915, 7.42647965860644167776315724949, 7.47906214811694943254418977697, 8.116653920015147863681030485165

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