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Properties

Label 4-41236-1.1-c1e2-0-0
Degree $4$
Conductor $41236$
Sign $1$
Analytic cond. $2.62924$
Root an. cond. $1.27337$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4-s − 3·9-s − 12-s + 5·13-s + 16-s − 3·17-s + 12·23-s − 25-s + 4·27-s + 3·29-s − 3·36-s − 5·39-s − 2·43-s − 48-s + 14·49-s + 3·51-s + 5·52-s − 9·53-s + 9·61-s + 64-s − 3·68-s − 12·69-s + 75-s + 16·79-s + 2·81-s − 3·87-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s − 9-s − 0.288·12-s + 1.38·13-s + 1/4·16-s − 0.727·17-s + 2.50·23-s − 1/5·25-s + 0.769·27-s + 0.557·29-s − 1/2·36-s − 0.800·39-s − 0.304·43-s − 0.144·48-s + 2·49-s + 0.420·51-s + 0.693·52-s − 1.23·53-s + 1.15·61-s + 1/8·64-s − 0.363·68-s − 1.44·69-s + 0.115·75-s + 1.80·79-s + 2/9·81-s − 0.321·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41236 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41236 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(41236\)    =    \(2^{2} \cdot 13^{2} \cdot 61\)
Sign: $1$
Analytic conductor: \(2.62924\)
Root analytic conductor: \(1.27337\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 41236,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.191018408\)
\(L(\frac12)\) \(\approx\) \(1.191018408\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_2$ \( 1 - 5 T + p T^{2} \)
61$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 8 T + p T^{2} ) \)
good3$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.b_e
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
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
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.19.a_q
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.23.am_de
29$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.ad_e
31$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.31.a_ack
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.37.a_abm
41$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \) 2.41.a_t
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.c_g
47$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.47.a_e
53$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.j_ec
59$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.59.a_abj
67$C_2^2$ \( 1 + 127 T^{2} + p^{2} T^{4} \) 2.67.a_ex
71$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.71.a_z
73$C_2^2$ \( 1 + 43 T^{2} + p^{2} T^{4} \) 2.73.a_br
79$C_2$$\times$$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.79.aq_fl
83$C_2^2$ \( 1 - 140 T^{2} + p^{2} T^{4} \) 2.83.a_afk
89$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.89.a_abm
97$C_2^2$ \( 1 + 121 T^{2} + p^{2} T^{4} \) 2.97.a_er
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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

−10.58837351913075848086010884932, −9.760182285447709251280546720851, −9.017616020046450196200111760464, −8.787536065726998351525549804983, −8.288318523746549799601056966921, −7.56593087837915052716736866948, −6.87177088419018989685114897789, −6.49093621243939814229473997917, −5.96516898549407527742924312255, −5.34198602984852888890532543752, −4.85045711571175414909871984758, −3.87437618396240466580640329320, −3.15354459997816701916536739942, −2.43415723323928784970438705373, −1.07123868476742935720148859681, 1.07123868476742935720148859681, 2.43415723323928784970438705373, 3.15354459997816701916536739942, 3.87437618396240466580640329320, 4.85045711571175414909871984758, 5.34198602984852888890532543752, 5.96516898549407527742924312255, 6.49093621243939814229473997917, 6.87177088419018989685114897789, 7.56593087837915052716736866948, 8.288318523746549799601056966921, 8.787536065726998351525549804983, 9.017616020046450196200111760464, 9.760182285447709251280546720851, 10.58837351913075848086010884932

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