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Properties

Label 4-8624e2-1.1-c1e2-0-3
Degree $4$
Conductor $74373376$
Sign $1$
Analytic cond. $4742.11$
Root an. cond. $8.29837$
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  − 3-s + 3·5-s − 9-s − 2·11-s + 2·13-s − 3·15-s + 4·17-s − 2·19-s + 7·23-s + 25-s − 4·29-s − 17·31-s + 2·33-s + 5·37-s − 2·39-s + 20·41-s − 8·43-s − 3·45-s + 4·47-s − 4·51-s + 8·53-s − 6·55-s + 2·57-s − 15·59-s − 8·61-s + 6·65-s + 67-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s − 1/3·9-s − 0.603·11-s + 0.554·13-s − 0.774·15-s + 0.970·17-s − 0.458·19-s + 1.45·23-s + 1/5·25-s − 0.742·29-s − 3.05·31-s + 0.348·33-s + 0.821·37-s − 0.320·39-s + 3.12·41-s − 1.21·43-s − 0.447·45-s + 0.583·47-s − 0.560·51-s + 1.09·53-s − 0.809·55-s + 0.264·57-s − 1.95·59-s − 1.02·61-s + 0.744·65-s + 0.122·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74373376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74373376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(74373376\)    =    \(2^{8} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(4742.11\)
Root analytic conductor: \(8.29837\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 74373376,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.629607573\)
\(L(\frac12)\) \(\approx\) \(1.629607573\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) 2.3.b_c
5$C_2^2$ \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.5.ad_i
13$C_4$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.13.ac_k
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
19$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_w
23$D_{4}$ \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.23.ah_cc
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$D_{4}$ \( 1 + 17 T + 130 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.31.r_fa
37$D_{4}$ \( 1 - 5 T + 76 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.37.af_cy
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.41.au_ha
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_be
53$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_cc
59$D_{4}$ \( 1 + 15 T + 170 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.59.p_go
61$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_cs
67$D_{4}$ \( 1 - T + 130 T^{2} - p T^{3} + p^{2} T^{4} \) 2.67.ab_fa
71$D_{4}$ \( 1 - 5 T + 110 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.71.af_eg
73$D_{4}$ \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_dq
79$D_{4}$ \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.79.c_fm
83$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.83.q_iw
89$D_{4}$ \( 1 - 21 T + 284 T^{2} - 21 p T^{3} + p^{2} T^{4} \) 2.89.av_ky
97$D_{4}$ \( 1 + 13 T + 232 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.97.n_iy
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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

−7.77338903999988000038602703280, −7.58938178859091446100141818697, −7.20494157532337757555080033874, −7.05304993261389219791783696386, −6.24681985257342302451242360293, −6.21807935631760219587492575700, −5.72126365569403913800766652732, −5.70168659769389723638079340182, −5.28275303428965737912242683834, −5.09634566216679401783634493761, −4.44085152021846122170457395216, −4.08815196912956055196668401363, −3.68963601465300729979918360750, −3.23281970372005801767444422986, −2.69208749093085038624165891354, −2.55162866280063207870795350546, −1.83063504038518845200777674797, −1.57918839345820614247782915445, −1.06271186926021691802176623021, −0.31640324540219889555662500084, 0.31640324540219889555662500084, 1.06271186926021691802176623021, 1.57918839345820614247782915445, 1.83063504038518845200777674797, 2.55162866280063207870795350546, 2.69208749093085038624165891354, 3.23281970372005801767444422986, 3.68963601465300729979918360750, 4.08815196912956055196668401363, 4.44085152021846122170457395216, 5.09634566216679401783634493761, 5.28275303428965737912242683834, 5.70168659769389723638079340182, 5.72126365569403913800766652732, 6.21807935631760219587492575700, 6.24681985257342302451242360293, 7.05304993261389219791783696386, 7.20494157532337757555080033874, 7.58938178859091446100141818697, 7.77338903999988000038602703280

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