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Properties

Label 4-2646e2-1.1-c1e2-0-11
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 − 4·8-s − 3·11-s + 2·13-s + 5·16-s + 3·17-s − 19-s + 6·22-s − 6·23-s + 5·25-s − 4·26-s + 6·29-s + 8·31-s − 6·32-s − 6·34-s + 4·37-s + 2·38-s − 9·41-s + 43-s − 9·44-s + 12·46-s − 12·47-s − 10·50-s + 6·52-s + 12·53-s − 12·58-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s − 0.904·11-s + 0.554·13-s + 5/4·16-s + 0.727·17-s − 0.229·19-s + 1.27·22-s − 1.25·23-s + 25-s − 0.784·26-s + 1.11·29-s + 1.43·31-s − 1.06·32-s − 1.02·34-s + 0.657·37-s + 0.324·38-s − 1.40·41-s + 0.152·43-s − 1.35·44-s + 1.76·46-s − 1.75·47-s − 1.41·50-s + 0.832·52-s + 1.64·53-s − 1.57·58-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

\(L(1)\) \(\approx\) \(1.215322309\)
\(L(\frac12)\) \(\approx\) \(1.215322309\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.5.a_af
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_ac
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.13.ac_aj
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.17.ad_ai
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.b_as
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_n
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_h
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2^2$ \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.37.ae_av
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.41.j_bo
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) 2.43.ab_abq
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.47.m_fa
53$C_2^2$ \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.53.am_dn
59$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.59.ag_ex
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.61.q_he
67$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.67.ak_gd
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2^2$ \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.73.al_bw
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.79.i_gs
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_cj
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.89.g_acb
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.af_acu
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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

−8.819306042559149366712982146397, −8.635057422957178197069485948737, −8.244534904495556194671033432831, −8.149078322124161705355530980907, −7.67802713463572480188146163166, −7.31805930353675071367748332626, −6.68750520259703005942461616312, −6.58086326082124629834983295199, −6.14458805906463879378489249533, −5.74136872007093498487334764758, −5.08160046829492507545557193059, −4.96318117081020586353393678419, −4.23683440606593792972655425284, −3.73511767660130153173940175456, −3.06181085702661795380378601278, −2.91505730668241754735582627927, −2.19041502823851050101064873308, −1.80330345108160920884028359169, −0.985264633737649693461087616249, −0.55812430525701541304521890626, 0.55812430525701541304521890626, 0.985264633737649693461087616249, 1.80330345108160920884028359169, 2.19041502823851050101064873308, 2.91505730668241754735582627927, 3.06181085702661795380378601278, 3.73511767660130153173940175456, 4.23683440606593792972655425284, 4.96318117081020586353393678419, 5.08160046829492507545557193059, 5.74136872007093498487334764758, 6.14458805906463879378489249533, 6.58086326082124629834983295199, 6.68750520259703005942461616312, 7.31805930353675071367748332626, 7.67802713463572480188146163166, 8.149078322124161705355530980907, 8.244534904495556194671033432831, 8.635057422957178197069485948737, 8.819306042559149366712982146397

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