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Properties

Label 4-2646e2-1.1-c1e2-0-18
Degree $4$
Conductor $7001316$
Sign $1$
Analytic cond. $446.409$
Root an. cond. $4.59656$
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 + 3·4-s + 5-s − 4·8-s − 2·10-s − 2·11-s − 2·13-s + 5·16-s + 7·19-s + 3·20-s + 4·22-s + 3·23-s + 5·25-s + 4·26-s − 8·29-s + 8·31-s − 6·32-s + 6·37-s − 14·38-s − 4·40-s − 12·41-s + 8·43-s − 6·44-s − 6·46-s + 16·47-s − 10·50-s − 6·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.447·5-s − 1.41·8-s − 0.632·10-s − 0.603·11-s − 0.554·13-s + 5/4·16-s + 1.60·19-s + 0.670·20-s + 0.852·22-s + 0.625·23-s + 25-s + 0.784·26-s − 1.48·29-s + 1.43·31-s − 1.06·32-s + 0.986·37-s − 2.27·38-s − 0.632·40-s − 1.87·41-s + 1.21·43-s − 0.904·44-s − 0.884·46-s + 2.33·47-s − 1.41·50-s − 0.832·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7001316\)    =    \(2^{2} \cdot 3^{6} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(446.409\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7001316,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.642915478\)
\(L(\frac12)\) \(\approx\) \(1.642915478\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) 2.5.ab_ae
11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_ah
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.c_aj
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.ah_be
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.23.ad_ao
29$C_2^2$ \( 1 + 8 T + 35 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.29.i_bj
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2^2$ \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_ab
41$C_2^2$ \( 1 + 12 T + 103 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.41.m_dz
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.43.ai_v
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$C_2^2$ \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.53.ae_abl
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \) 2.61.aba_lf
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.67.e_fi
71$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.71.ak_gl
73$C_2^2$ \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.73.ao_et
79$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.79.w_kt
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_cj
89$C_2^2$ \( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.89.ao_ed
97$C_2^2$ \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.97.ac_adp
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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

−9.081625475546759282827077991691, −8.826585285347409452195513090994, −8.132468155162315697915566425840, −8.114375245403130292014712632340, −7.42742179468375629203185932962, −7.41395745203237461966689254447, −6.83366045478317561189287639576, −6.64275848245892408697667163400, −6.03643769234622137262394997516, −5.56355092750483524052143101460, −5.21999303492029263829623498340, −5.05587892203439246336886191609, −4.18165203317382792790840448651, −3.79645442311654491715841317564, −2.97130776362987681804488822422, −2.85909563204315196840018581742, −2.27343109450919242230066159046, −1.80071500542722033274808889208, −0.887805318346728249203421700831, −0.71761287465178612011908821990, 0.71761287465178612011908821990, 0.887805318346728249203421700831, 1.80071500542722033274808889208, 2.27343109450919242230066159046, 2.85909563204315196840018581742, 2.97130776362987681804488822422, 3.79645442311654491715841317564, 4.18165203317382792790840448651, 5.05587892203439246336886191609, 5.21999303492029263829623498340, 5.56355092750483524052143101460, 6.03643769234622137262394997516, 6.64275848245892408697667163400, 6.83366045478317561189287639576, 7.41395745203237461966689254447, 7.42742179468375629203185932962, 8.114375245403130292014712632340, 8.132468155162315697915566425840, 8.826585285347409452195513090994, 9.081625475546759282827077991691

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