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Properties

Label 4-325e2-1.1-c1e2-0-21
Degree $4$
Conductor $105625$
Sign $1$
Analytic cond. $6.73474$
Root an. cond. $1.61094$
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  + 4·3-s + 3·4-s + 6·9-s + 12·12-s + 6·13-s + 5·16-s − 10·17-s − 8·23-s − 4·27-s − 2·29-s + 18·36-s + 24·39-s + 8·43-s + 20·48-s − 11·49-s − 40·51-s + 18·52-s + 6·53-s + 2·61-s + 3·64-s − 30·68-s − 32·69-s + 20·79-s − 37·81-s − 8·87-s − 24·92-s + 2·101-s + ⋯
L(s)  = 1  + 2.30·3-s + 3/2·4-s + 2·9-s + 3.46·12-s + 1.66·13-s + 5/4·16-s − 2.42·17-s − 1.66·23-s − 0.769·27-s − 0.371·29-s + 3·36-s + 3.84·39-s + 1.21·43-s + 2.88·48-s − 1.57·49-s − 5.60·51-s + 2.49·52-s + 0.824·53-s + 0.256·61-s + 3/8·64-s − 3.63·68-s − 3.85·69-s + 2.25·79-s − 4.11·81-s − 0.857·87-s − 2.50·92-s + 0.199·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(105625\)    =    \(5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(6.73474\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 105625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.359759685\)
\(L(\frac12)\) \(\approx\) \(4.359759685\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad5 \( 1 \)
13$C_2$ \( 1 - 6 T + p T^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.2.a_ad
3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.3.ae_k
7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.7.a_l
11$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.11.a_an
17$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.17.k_ch
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.19.a_aw
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.23.i_ck
29$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.29.c_ch
31$C_2^2$ \( 1 - 61 T^{2} + p^{2} T^{4} \) 2.31.a_acj
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.37.a_acg
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.a_as
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \) 2.47.a_abt
53$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.53.ag_el
59$C_2^2$ \( 1 - 109 T^{2} + p^{2} T^{4} \) 2.59.a_aef
61$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.61.ac_et
67$C_2^2$ \( 1 - 125 T^{2} + p^{2} T^{4} \) 2.67.a_aev
71$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.71.a_ada
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.73.a_afa
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.79.au_jy
83$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \) 2.83.a_adh
89$C_2^2$ \( 1 + 146 T^{2} + p^{2} T^{4} \) 2.89.a_fq
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.97.a_c
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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

−11.53750914130824073006715677949, −11.46700607029575927390978402793, −10.87292610258284163388843425124, −10.61442414413858874068319526456, −9.851308563878474578219364909445, −9.242503401649232239745203316592, −9.007539828240496874666194551557, −8.483951197797741320423570850172, −8.093642427178320429863755108476, −7.81268210324789693045844564845, −7.12496940881709433883397922404, −6.60602589636245882480353366966, −6.14417059267584349737480582605, −5.70063770840894105669236331940, −4.45292740883216667540377214807, −3.84983222341498955142917010222, −3.42556616076850966040137692301, −2.71392006719634239188808586349, −2.10448523742543851980756786708, −1.88414572462430415752265888102, 1.88414572462430415752265888102, 2.10448523742543851980756786708, 2.71392006719634239188808586349, 3.42556616076850966040137692301, 3.84983222341498955142917010222, 4.45292740883216667540377214807, 5.70063770840894105669236331940, 6.14417059267584349737480582605, 6.60602589636245882480353366966, 7.12496940881709433883397922404, 7.81268210324789693045844564845, 8.093642427178320429863755108476, 8.483951197797741320423570850172, 9.007539828240496874666194551557, 9.242503401649232239745203316592, 9.851308563878474578219364909445, 10.61442414413858874068319526456, 10.87292610258284163388843425124, 11.46700607029575927390978402793, 11.53750914130824073006715677949

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