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Properties

Label 4-5415e2-1.1-c1e2-0-5
Degree $4$
Conductor $29322225$
Sign $1$
Analytic cond. $1869.61$
Root an. cond. $6.57563$
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  + 2·3-s + 3·4-s + 2·5-s − 2·7-s + 3·9-s + 6·11-s + 6·12-s + 6·13-s + 4·15-s + 5·16-s − 8·17-s + 6·20-s − 4·21-s + 8·23-s + 3·25-s + 4·27-s − 6·28-s − 2·29-s − 12·31-s + 12·33-s − 4·35-s + 9·36-s − 2·37-s + 12·39-s + 14·41-s + 6·43-s + 18·44-s + ⋯
L(s)  = 1  + 1.15·3-s + 3/2·4-s + 0.894·5-s − 0.755·7-s + 9-s + 1.80·11-s + 1.73·12-s + 1.66·13-s + 1.03·15-s + 5/4·16-s − 1.94·17-s + 1.34·20-s − 0.872·21-s + 1.66·23-s + 3/5·25-s + 0.769·27-s − 1.13·28-s − 0.371·29-s − 2.15·31-s + 2.08·33-s − 0.676·35-s + 3/2·36-s − 0.328·37-s + 1.92·39-s + 2.18·41-s + 0.914·43-s + 2.71·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29322225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29322225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(29322225\)    =    \(3^{2} \cdot 5^{2} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(1869.61\)
Root analytic conductor: \(6.57563\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 29322225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(11.42037005\)
\(L(\frac12)\) \(\approx\) \(11.42037005\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
19 \( 1 \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.2.a_ad
7$D_{4}$ \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_i
11$D_{4}$ \( 1 - 6 T + 24 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_y
13$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.13.ag_bc
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.17.i_by
23$D_{4}$ \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.23.ai_bi
29$D_{4}$ \( 1 + 2 T - 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.29.c_ae
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.31.m_du
37$D_{4}$ \( 1 + 2 T + 68 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.37.c_cq
41$D_{4}$ \( 1 - 14 T + 124 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.41.ao_eu
43$D_{4}$ \( 1 - 6 T + 88 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.43.ag_dk
47$D_{4}$ \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.47.ai_de
53$D_{4}$ \( 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.53.m_ek
59$D_{4}$ \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_ew
61$D_{4}$ \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.61.m_fa
67$D_{4}$ \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_bm
71$D_{4}$ \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.71.e_eo
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.73.au_jm
79$D_{4}$ \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.79.ai_ck
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.83.am_hu
89$D_{4}$ \( 1 - 18 T + 196 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.89.as_ho
97$D_{4}$ \( 1 + 10 T + 156 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.97.k_ga
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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

−8.577154902594789531918917152613, −7.73628304132250010195559367429, −7.60066817409623747775818506043, −7.34446391802191419039812226846, −6.71265043129274183556570302753, −6.51981573498064807130381809168, −6.46841948018535075575916695413, −5.92680962740223158359931565238, −5.88487183819479583497450111514, −5.02467014561362178787827386304, −4.62309372226598258051105854047, −4.09839966178114707604350028136, −3.71670326153197024388569019651, −3.40874528039129945231204597838, −3.04594607498498557803039185198, −2.51850215320828028210424600792, −2.17789635837277201870717159396, −1.59273821173736288635935958014, −1.51476721695981464304598333712, −0.77688744277918160687460269859, 0.77688744277918160687460269859, 1.51476721695981464304598333712, 1.59273821173736288635935958014, 2.17789635837277201870717159396, 2.51850215320828028210424600792, 3.04594607498498557803039185198, 3.40874528039129945231204597838, 3.71670326153197024388569019651, 4.09839966178114707604350028136, 4.62309372226598258051105854047, 5.02467014561362178787827386304, 5.88487183819479583497450111514, 5.92680962740223158359931565238, 6.46841948018535075575916695413, 6.51981573498064807130381809168, 6.71265043129274183556570302753, 7.34446391802191419039812226846, 7.60066817409623747775818506043, 7.73628304132250010195559367429, 8.577154902594789531918917152613

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