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Properties

Label 4-425e2-1.1-c1e2-0-7
Degree $4$
Conductor $180625$
Sign $1$
Analytic cond. $11.5168$
Root an. cond. $1.84218$
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 + 6·7-s + 2·9-s − 6·11-s + 6·12-s + 5·16-s − 8·17-s + 12·21-s + 2·23-s + 6·27-s + 18·28-s + 6·29-s − 2·31-s − 12·33-s + 6·36-s + 6·37-s − 6·41-s − 18·44-s − 4·47-s + 10·48-s + 18·49-s − 16·51-s + 2·61-s + 12·63-s + 3·64-s − 12·67-s + ⋯
L(s)  = 1  + 1.15·3-s + 3/2·4-s + 2.26·7-s + 2/3·9-s − 1.80·11-s + 1.73·12-s + 5/4·16-s − 1.94·17-s + 2.61·21-s + 0.417·23-s + 1.15·27-s + 3.40·28-s + 1.11·29-s − 0.359·31-s − 2.08·33-s + 36-s + 0.986·37-s − 0.937·41-s − 2.71·44-s − 0.583·47-s + 1.44·48-s + 18/7·49-s − 2.24·51-s + 0.256·61-s + 1.51·63-s + 3/8·64-s − 1.46·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(180625\)    =    \(5^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(11.5168\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 180625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.131729909\)
\(L(\frac12)\) \(\approx\) \(4.131729909\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17$C_2$ \( 1 + 8 T + p T^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.2.a_ad
3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_c
7$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.7.ag_s
11$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.11.g_s
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.23.ac_c
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.ag_s
31$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.31.c_c
37$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_s
41$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_s
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.43.a_cg
47$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.47.e_du
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.53.a_ady
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.59.a_ade
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.ac_c
67$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.67.m_go
71$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.71.ag_s
73$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.73.g_s
79$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.79.ao_du
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \) 2.83.a_afu
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.97.ag_s
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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

−11.19893813434124491607922070241, −10.88645862620098036861847931077, −10.77482184493561659314772457450, −10.26654997819384697253324179092, −9.615468957415970625998706701394, −8.873514114358976013800230556552, −8.434368629218127139620358629963, −8.270817963199271735754877227321, −7.71397706421828913638312442505, −7.50388973240751665204042357925, −6.78889168013275970491057941218, −6.49956799124622307315373326011, −5.61144542850559222660657547199, −4.93285456194737219430732325573, −4.76534627859259058160597149231, −4.02133466765917442898045466286, −2.80570019809810177639246574631, −2.71197284432485808348611293621, −2.07086679002784255870595377740, −1.48583317434370583270727717089, 1.48583317434370583270727717089, 2.07086679002784255870595377740, 2.71197284432485808348611293621, 2.80570019809810177639246574631, 4.02133466765917442898045466286, 4.76534627859259058160597149231, 4.93285456194737219430732325573, 5.61144542850559222660657547199, 6.49956799124622307315373326011, 6.78889168013275970491057941218, 7.50388973240751665204042357925, 7.71397706421828913638312442505, 8.270817963199271735754877227321, 8.434368629218127139620358629963, 8.873514114358976013800230556552, 9.615468957415970625998706701394, 10.26654997819384697253324179092, 10.77482184493561659314772457450, 10.88645862620098036861847931077, 11.19893813434124491607922070241

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