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Properties

Label 4-546e2-1.1-c1e2-0-12
Degree $4$
Conductor $298116$
Sign $1$
Analytic cond. $19.0081$
Root an. cond. $2.08802$
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  + 4-s − 2·7-s − 3·9-s − 2·13-s + 16-s − 6·25-s − 2·28-s − 16·31-s − 3·36-s + 12·37-s + 8·43-s + 3·49-s − 2·52-s + 20·61-s + 6·63-s + 64-s + 8·67-s + 4·73-s + 16·79-s + 9·81-s + 4·91-s + 4·97-s − 6·100-s + 32·103-s − 4·109-s − 2·112-s + 6·117-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.755·7-s − 9-s − 0.554·13-s + 1/4·16-s − 6/5·25-s − 0.377·28-s − 2.87·31-s − 1/2·36-s + 1.97·37-s + 1.21·43-s + 3/7·49-s − 0.277·52-s + 2.56·61-s + 0.755·63-s + 1/8·64-s + 0.977·67-s + 0.468·73-s + 1.80·79-s + 81-s + 0.419·91-s + 0.406·97-s − 3/5·100-s + 3.15·103-s − 0.383·109-s − 0.188·112-s + 0.554·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(298116\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(19.0081\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 298116,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.270689638\)
\(L(\frac12)\) \(\approx\) \(1.270689638\)
\(L(\frac{3}{2})\) 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$ \( ( 1 - T )( 1 + T ) \)
3$C_2$ \( 1 + p T^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.a_as
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.37.am_eg
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.a_cc
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.61.au_io
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.a_da
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.89.a_afq
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

−8.866721919717987183503254472214, −8.369588935348609600456733576104, −7.80088849136349478139956350243, −7.40803372522333751876994487224, −7.08827055361374002488782088399, −6.19483789697770666814563584573, −6.19217622188912966190136871625, −5.44024157926880829299881771792, −5.21535057877927416914375242112, −4.23885567109427787805557130576, −3.67764363944170895713490289621, −3.27331439299451669391457686232, −2.29694658439320588323627036273, −2.18713180950649411226712364496, −0.62603155646423681877598241271, 0.62603155646423681877598241271, 2.18713180950649411226712364496, 2.29694658439320588323627036273, 3.27331439299451669391457686232, 3.67764363944170895713490289621, 4.23885567109427787805557130576, 5.21535057877927416914375242112, 5.44024157926880829299881771792, 6.19217622188912966190136871625, 6.19483789697770666814563584573, 7.08827055361374002488782088399, 7.40803372522333751876994487224, 7.80088849136349478139956350243, 8.369588935348609600456733576104, 8.866721919717987183503254472214

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