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Properties

Label 4-65e4-1.1-c1e2-0-7
Degree $4$
Conductor $17850625$
Sign $1$
Analytic cond. $1138.17$
Root an. cond. $5.80833$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·4-s + 4·7-s − 3·8-s − 9-s − 4·11-s + 4·14-s + 16-s − 2·17-s − 18-s − 4·19-s − 4·22-s − 12·23-s − 8·28-s − 6·29-s + 2·32-s − 2·34-s + 2·36-s + 6·37-s − 4·38-s − 6·41-s + 8·43-s + 8·44-s − 12·46-s + 8·47-s + 3·49-s − 12·53-s + ⋯
L(s)  = 1  + 0.707·2-s − 4-s + 1.51·7-s − 1.06·8-s − 1/3·9-s − 1.20·11-s + 1.06·14-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.852·22-s − 2.50·23-s − 1.51·28-s − 1.11·29-s + 0.353·32-s − 0.342·34-s + 1/3·36-s + 0.986·37-s − 0.648·38-s − 0.937·41-s + 1.21·43-s + 1.20·44-s − 1.76·46-s + 1.16·47-s + 3/7·49-s − 1.64·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17850625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(17850625\)    =    \(5^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(1138.17\)
Root analytic conductor: \(5.80833\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 17850625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) 2.2.ab_d
3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
7$D_{4}$ \( 1 - 4 T + 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.7.ae_n
11$D_{4}$ \( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_v
17$D_{4}$ \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.17.c_p
19$D_{4}$ \( 1 + 4 T + 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_bl
23$D_{4}$ \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.23.m_cz
29$D_{4}$ \( 1 + 6 T + 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.29.g_bv
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.37.ag_df
41$D_{4}$ \( 1 + 6 T + 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_l
43$D_{4}$ \( 1 - 8 T + 97 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.43.ai_dt
47$D_{4}$ \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.47.ai_be
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$D_{4}$ \( 1 + 12 T + 109 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_ef
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.61.ac_acf
67$D_{4}$ \( 1 - 8 T + 105 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_eb
71$D_{4}$ \( 1 - 8 T + 153 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.71.ai_fx
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.83.a_di
89$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.89.as_jz
97$D_{4}$ \( 1 - 2 T + 175 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.97.ac_gt
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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

−8.087234557150305884667686989639, −7.958229628832409511094873570826, −7.49913649058071292493027509767, −7.49551736205903165361606262630, −6.48203505770921549965323325304, −6.30394114127740485426982859692, −5.81080139285793303657872092986, −5.57730227759565892419731106937, −5.15356398115281101307829039802, −4.75111956772124458897081252200, −4.48133737092415327170972899042, −4.23323535581262617800236990107, −3.72641974028131388981094530063, −3.39145522878040781070331533016, −2.63092035701335290833138503604, −2.11653304260759616848007621675, −1.99151492545537500123227451920, −1.14727825542298722210467547106, 0, 0, 1.14727825542298722210467547106, 1.99151492545537500123227451920, 2.11653304260759616848007621675, 2.63092035701335290833138503604, 3.39145522878040781070331533016, 3.72641974028131388981094530063, 4.23323535581262617800236990107, 4.48133737092415327170972899042, 4.75111956772124458897081252200, 5.15356398115281101307829039802, 5.57730227759565892419731106937, 5.81080139285793303657872092986, 6.30394114127740485426982859692, 6.48203505770921549965323325304, 7.49551736205903165361606262630, 7.49913649058071292493027509767, 7.958229628832409511094873570826, 8.087234557150305884667686989639

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