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Properties

Label 4-8349e2-1.1-c1e2-0-1
Degree $4$
Conductor $69705801$
Sign $1$
Analytic cond. $4444.50$
Root an. cond. $8.16499$
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 − 4-s − 6·5-s + 3·9-s + 2·12-s + 12·15-s − 3·16-s + 6·20-s + 2·23-s + 17·25-s − 4·27-s + 8·31-s − 3·36-s + 8·37-s − 18·45-s + 6·48-s + 13·49-s − 18·53-s + 18·59-s − 12·60-s + 7·64-s + 26·67-s − 4·69-s + 18·71-s − 34·75-s + 18·80-s + 5·81-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s − 2.68·5-s + 9-s + 0.577·12-s + 3.09·15-s − 3/4·16-s + 1.34·20-s + 0.417·23-s + 17/5·25-s − 0.769·27-s + 1.43·31-s − 1/2·36-s + 1.31·37-s − 2.68·45-s + 0.866·48-s + 13/7·49-s − 2.47·53-s + 2.34·59-s − 1.54·60-s + 7/8·64-s + 3.17·67-s − 0.481·69-s + 2.13·71-s − 3.92·75-s + 2.01·80-s + 5/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69705801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69705801 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(69705801\)    =    \(3^{2} \cdot 11^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(4444.50\)
Root analytic conductor: \(8.16499\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 69705801,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7558008922\)
\(L(\frac12)\) \(\approx\) \(0.7558008922\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
23$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.2.a_b
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.5.g_t
7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.7.a_an
13$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \) 2.13.a_x
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.17.a_ao
19$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \) 2.19.a_bj
29$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.29.a_k
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.37.ai_dm
41$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.41.a_aba
43$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.43.a_cw
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.53.s_hf
59$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.59.as_hr
61$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.61.a_es
67$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \) 2.67.aba_lr
71$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.71.as_ip
73$C_2^2$ \( 1 + 143 T^{2} + p^{2} T^{4} \) 2.73.a_fn
79$C_2^2$ \( 1 + 83 T^{2} + p^{2} T^{4} \) 2.79.a_df
83$C_2^2$ \( 1 - 134 T^{2} + p^{2} T^{4} \) 2.83.a_afe
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.97.aq_jy
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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

−7.79369030209787245760599536774, −7.75938630665330189228983193228, −7.31227841084767205540134030670, −6.90087654052754902736493526701, −6.64365935279650456273341152873, −6.43142681036308275889569540450, −5.94949984786149832297025335718, −5.28092747176269445710006847732, −5.10765369181472608697138319425, −4.89728108244230320461208912159, −4.32531250203750995638951849772, −4.03496162580471825279748107260, −3.91403328624653989304455113290, −3.65793350539223991284238455326, −2.81182889064679677112178949236, −2.68716809064709041360342941713, −1.95104083942244091599974349277, −1.09320895185131796979173745916, −0.60583176823298959623357514959, −0.45193842986506214198940235221, 0.45193842986506214198940235221, 0.60583176823298959623357514959, 1.09320895185131796979173745916, 1.95104083942244091599974349277, 2.68716809064709041360342941713, 2.81182889064679677112178949236, 3.65793350539223991284238455326, 3.91403328624653989304455113290, 4.03496162580471825279748107260, 4.32531250203750995638951849772, 4.89728108244230320461208912159, 5.10765369181472608697138319425, 5.28092747176269445710006847732, 5.94949984786149832297025335718, 6.43142681036308275889569540450, 6.64365935279650456273341152873, 6.90087654052754902736493526701, 7.31227841084767205540134030670, 7.75938630665330189228983193228, 7.79369030209787245760599536774

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