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Properties

Label 4-153e2-1.1-c1e2-0-1
Degree $4$
Conductor $23409$
Sign $1$
Analytic cond. $1.49257$
Root an. cond. $1.10531$
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  − 4·4-s − 2·13-s + 12·16-s + 10·19-s − 7·25-s + 22·43-s + 14·49-s + 8·52-s − 32·64-s + 16·67-s − 40·76-s + 28·100-s − 38·103-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 23·169-s − 88·172-s + 173-s + 179-s + ⋯
L(s)  = 1  − 2·4-s − 0.554·13-s + 3·16-s + 2.29·19-s − 7/5·25-s + 3.35·43-s + 2·49-s + 1.10·52-s − 4·64-s + 1.95·67-s − 4.58·76-s + 14/5·100-s − 3.74·103-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.76·169-s − 6.70·172-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23409 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23409 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(23409\)    =    \(3^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(1.49257\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 23409,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7261324924\)
\(L(\frac12)\) \(\approx\) \(0.7261324924\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17$C_2$ \( 1 + p T^{2} \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.2.a_e
5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.5.a_h
7$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.7.a_ao
11$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.11.a_af
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.13.c_bb
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.19.ak_cl
23$C_2^2$ \( 1 - 29 T^{2} + p^{2} T^{4} \) 2.23.a_abd
29$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.29.a_k
31$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.31.a_ack
37$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.37.a_acw
41$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \) 2.41.a_acn
43$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.43.aw_hz
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.61.a_aes
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.67.aq_hq
71$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \) 2.71.a_fa
73$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.73.a_afq
79$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.79.a_agc
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.97.a_ahm
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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

−13.53775892560318106310910398985, −12.73771041920417490052266237753, −12.15041026925029490552105856456, −12.12811474436563540864673653366, −11.20302199190088921731449458790, −10.60469007478186818262450510920, −9.894078556665582167550913173828, −9.603643811261349093777838647689, −9.262694930711212273657472698289, −8.756006203078045934159160170532, −7.906924833504903118323478670203, −7.72407406514636345892186587754, −7.07083890770692774176396054804, −5.77231813928881131795249053068, −5.58995675492137965916450338420, −4.97419961464472443978675598258, −4.11086895909721362941669626921, −3.79280369284563752828600486905, −2.72129187565145507994058986416, −0.931369063223970506871556425315, 0.931369063223970506871556425315, 2.72129187565145507994058986416, 3.79280369284563752828600486905, 4.11086895909721362941669626921, 4.97419961464472443978675598258, 5.58995675492137965916450338420, 5.77231813928881131795249053068, 7.07083890770692774176396054804, 7.72407406514636345892186587754, 7.906924833504903118323478670203, 8.756006203078045934159160170532, 9.262694930711212273657472698289, 9.603643811261349093777838647689, 9.894078556665582167550913173828, 10.60469007478186818262450510920, 11.20302199190088921731449458790, 12.12811474436563540864673653366, 12.15041026925029490552105856456, 12.73771041920417490052266237753, 13.53775892560318106310910398985

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