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Properties

Label 4-364e2-1.1-c1e2-0-6
Degree $4$
Conductor $132496$
Sign $1$
Analytic cond. $8.44805$
Root an. cond. $1.70486$
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  + 3·3-s + 3·5-s − 4·7-s + 3·9-s + 5·11-s − 2·13-s + 9·15-s + 17-s + 7·19-s − 12·21-s + 23-s + 5·25-s − 4·29-s + 5·31-s + 15·33-s − 12·35-s + 11·37-s − 6·39-s − 12·41-s − 24·43-s + 9·45-s + 7·47-s + 9·49-s + 3·51-s − 11·53-s + 15·55-s + 21·57-s + ⋯
L(s)  = 1  + 1.73·3-s + 1.34·5-s − 1.51·7-s + 9-s + 1.50·11-s − 0.554·13-s + 2.32·15-s + 0.242·17-s + 1.60·19-s − 2.61·21-s + 0.208·23-s + 25-s − 0.742·29-s + 0.898·31-s + 2.61·33-s − 2.02·35-s + 1.80·37-s − 0.960·39-s − 1.87·41-s − 3.65·43-s + 1.34·45-s + 1.02·47-s + 9/7·49-s + 0.420·51-s − 1.51·53-s + 2.02·55-s + 2.78·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(132496\)    =    \(2^{4} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(8.44805\)
Root analytic conductor: \(1.70486\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 132496,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.322551804\)
\(L(\frac12)\) \(\approx\) \(3.322551804\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) 2.3.ad_g
5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.5.ad_e
11$C_2^2$ \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.11.af_o
17$C_2^2$ \( 1 - T - 16 T^{2} - p T^{3} + p^{2} T^{4} \) 2.17.ab_aq
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.ah_be
23$C_2^2$ \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) 2.23.ab_aw
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2^2$ \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.31.af_ag
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.37.al_dg
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.43.y_iw
47$C_2^2$ \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.47.ah_c
53$C_2^2$ \( 1 + 11 T + 68 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.53.l_cq
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.59.d_aby
61$C_2^2$ \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.61.af_abk
67$C_2^2$ \( 1 - T - 66 T^{2} - p T^{3} + p^{2} T^{4} \) 2.67.ab_aco
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2^2$ \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.73.al_bw
79$C_2^2$ \( 1 - 5 T - 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.79.af_acc
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.83.ai_ha
89$C_2^2$ \( 1 + 5 T - 64 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.89.f_acm
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−11.60967143954135412666047246861, −11.39506752579949328730911537630, −10.19472842374802337302045069941, −10.16913671021276343889255111225, −9.491743214845896128690214682996, −9.403657193569161283468588170662, −9.239411569961367696988309283461, −8.398323846673456380759246482547, −8.187036073048470485208941784715, −7.39511532184473414310945453069, −6.83745258258507834535094057054, −6.46422035105171359304400520942, −6.06715735369941119631466392143, −5.27722160762044589401332269336, −4.75264551703487527135149599897, −3.52704762417780248294167260563, −3.47753579934201837203442846534, −2.86655638765293398172969327486, −2.19248755052065571957693200249, −1.33026426814471304560485658599, 1.33026426814471304560485658599, 2.19248755052065571957693200249, 2.86655638765293398172969327486, 3.47753579934201837203442846534, 3.52704762417780248294167260563, 4.75264551703487527135149599897, 5.27722160762044589401332269336, 6.06715735369941119631466392143, 6.46422035105171359304400520942, 6.83745258258507834535094057054, 7.39511532184473414310945453069, 8.187036073048470485208941784715, 8.398323846673456380759246482547, 9.239411569961367696988309283461, 9.403657193569161283468588170662, 9.491743214845896128690214682996, 10.16913671021276343889255111225, 10.19472842374802337302045069941, 11.39506752579949328730911537630, 11.60967143954135412666047246861

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