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Properties

Label 4-325e2-1.1-c1e2-0-8
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  − 2·2-s − 4-s + 10·7-s + 8·8-s + 2·9-s − 4·13-s − 20·14-s − 7·16-s − 4·18-s + 8·26-s − 10·28-s + 2·29-s − 14·32-s − 2·36-s − 8·37-s + 14·47-s + 61·49-s + 4·52-s + 80·56-s − 4·58-s + 2·61-s + 20·63-s + 35·64-s − 6·67-s + 16·72-s + 8·73-s + 16·74-s + ⋯
L(s)  = 1  − 1.41·2-s − 1/2·4-s + 3.77·7-s + 2.82·8-s + 2/3·9-s − 1.10·13-s − 5.34·14-s − 7/4·16-s − 0.942·18-s + 1.56·26-s − 1.88·28-s + 0.371·29-s − 2.47·32-s − 1/3·36-s − 1.31·37-s + 2.04·47-s + 61/7·49-s + 0.554·52-s + 10.6·56-s − 0.525·58-s + 0.256·61-s + 2.51·63-s + 35/8·64-s − 0.733·67-s + 1.88·72-s + 0.936·73-s + 1.85·74-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

\(L(1)\) \(\approx\) \(0.9128531535\)
\(L(\frac12)\) \(\approx\) \(0.9128531535\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 4 T + p T^{2} \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.2.c_f
3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
7$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.7.ak_bn
11$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.11.a_an
17$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \) 2.17.a_aj
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.19.a_aw
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.29.ac_ch
31$C_2^2$ \( 1 - 61 T^{2} + p^{2} T^{4} \) 2.31.a_acj
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.37.i_dm
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.a_as
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.43.a_acs
47$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.47.ao_fn
53$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \) 2.53.a_adt
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$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.67.g_fn
71$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.71.a_ada
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.73.ai_gg
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.79.u_jy
83$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.83.s_jn
89$C_2^2$ \( 1 + 146 T^{2} + p^{2} T^{4} \) 2.89.a_fq
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.97.abc_pa
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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

−11.46734096261164614453853057994, −11.38502158068163916495467263841, −10.63364356860836270969075467070, −10.42012512599753505814830928101, −10.09834917257210647979140745822, −9.391692171468682841177486098011, −8.803311490432469805165097172784, −8.498160332307600170901938960037, −8.350847253418280992659148680446, −7.55906263091254102720028666402, −7.44142842120530832435795620294, −7.20677905564976580363652819523, −5.59811001088471801996347295521, −5.24727016759936878409990803024, −4.60124737174930514912143557880, −4.57814842204522320964446182267, −3.98537530070758719262941598905, −2.20372613703075722135017250404, −1.64390557452230311934047307962, −1.03497245506912373904505206560, 1.03497245506912373904505206560, 1.64390557452230311934047307962, 2.20372613703075722135017250404, 3.98537530070758719262941598905, 4.57814842204522320964446182267, 4.60124737174930514912143557880, 5.24727016759936878409990803024, 5.59811001088471801996347295521, 7.20677905564976580363652819523, 7.44142842120530832435795620294, 7.55906263091254102720028666402, 8.350847253418280992659148680446, 8.498160332307600170901938960037, 8.803311490432469805165097172784, 9.391692171468682841177486098011, 10.09834917257210647979140745822, 10.42012512599753505814830928101, 10.63364356860836270969075467070, 11.38502158068163916495467263841, 11.46734096261164614453853057994

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