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Properties

Label 4-7605e2-1.1-c1e2-0-7
Degree $4$
Conductor $57836025$
Sign $1$
Analytic cond. $3687.67$
Root an. cond. $7.79270$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 2·5-s − 3·7-s − 3·8-s + 2·10-s − 7·11-s + 3·14-s + 16-s − 5·17-s + 6·19-s − 2·20-s + 7·22-s + 7·23-s + 3·25-s − 3·28-s + 12·31-s + 32-s + 5·34-s + 6·35-s + 7·37-s − 6·38-s + 6·40-s − 5·41-s − 6·43-s − 7·44-s − 7·46-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.894·5-s − 1.13·7-s − 1.06·8-s + 0.632·10-s − 2.11·11-s + 0.801·14-s + 1/4·16-s − 1.21·17-s + 1.37·19-s − 0.447·20-s + 1.49·22-s + 1.45·23-s + 3/5·25-s − 0.566·28-s + 2.15·31-s + 0.176·32-s + 0.857·34-s + 1.01·35-s + 1.15·37-s − 0.973·38-s + 0.948·40-s − 0.780·41-s − 0.914·43-s − 1.05·44-s − 1.03·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57836025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57836025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(57836025\)    =    \(3^{4} \cdot 5^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(3687.67\)
Root analytic conductor: \(7.79270\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 57836025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
13 \( 1 \)
good2$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \) 2.2.b_a
7$D_{4}$ \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.7.d_m
11$D_{4}$ \( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.11.h_be
17$D_{4}$ \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.17.f_bk
19$D_{4}$ \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.19.ag_be
23$D_{4}$ \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.23.ah_cc
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.31.am_du
37$D_{4}$ \( 1 - 7 T + 82 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.37.ah_de
41$D_{4}$ \( 1 + 5 T + 84 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.41.f_dg
43$D_{4}$ \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.43.g_da
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.47.ai_eg
53$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \) 2.53.b_a
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$D_{4}$ \( 1 + 21 T + 228 T^{2} + 21 p T^{3} + p^{2} T^{4} \) 2.61.v_iu
67$D_{4}$ \( 1 + 14 T + 166 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.67.o_gk
71$D_{4}$ \( 1 - 5 T + 110 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.71.af_eg
73$D_{4}$ \( 1 - 14 T + 178 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.73.ao_gw
79$D_{4}$ \( 1 + 7 T + 166 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.79.h_gk
83$D_{4}$ \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.83.ac_o
89$D_{4}$ \( 1 + 15 T + 196 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.89.p_ho
97$D_{4}$ \( 1 - T - 14 T^{2} - p T^{3} + p^{2} T^{4} \) 2.97.ab_ao
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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

−7.66035492168789722080708926758, −7.44678258649630959482507273818, −7.12605822728158147997025266434, −6.63964696638233355645423562814, −6.47082801856555489994885442229, −6.06866708255272678559196993435, −5.75903164891355035481811542152, −5.15631839118667501317859348354, −4.79852667983541498008225489309, −4.70175760100627288524333752427, −4.14429745540867689254557136902, −3.50259729499955879015198384677, −3.02396181366965759050994627630, −2.83262970091072605425454747745, −2.82484777050137011511332243547, −2.17181526986574336076866754725, −1.33444030102530712724393804469, −0.78096296652164771620961857400, 0, 0, 0.78096296652164771620961857400, 1.33444030102530712724393804469, 2.17181526986574336076866754725, 2.82484777050137011511332243547, 2.83262970091072605425454747745, 3.02396181366965759050994627630, 3.50259729499955879015198384677, 4.14429745540867689254557136902, 4.70175760100627288524333752427, 4.79852667983541498008225489309, 5.15631839118667501317859348354, 5.75903164891355035481811542152, 6.06866708255272678559196993435, 6.47082801856555489994885442229, 6.63964696638233355645423562814, 7.12605822728158147997025266434, 7.44678258649630959482507273818, 7.66035492168789722080708926758

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