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Properties

Label 4-546e2-1.1-c1e2-0-18
Degree $4$
Conductor $298116$
Sign $1$
Analytic cond. $19.0081$
Root an. cond. $2.08802$
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 − 2·9-s + 12-s − 4·13-s + 16-s − 5·17-s + 10·23-s + 4·25-s − 5·27-s + 5·29-s − 2·36-s − 4·39-s + 13·43-s + 48-s − 49-s − 5·51-s − 4·52-s − 5·53-s + 11·61-s + 64-s − 5·68-s + 10·69-s + 4·75-s + 15·79-s + 81-s + 5·87-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s − 2/3·9-s + 0.288·12-s − 1.10·13-s + 1/4·16-s − 1.21·17-s + 2.08·23-s + 4/5·25-s − 0.962·27-s + 0.928·29-s − 1/3·36-s − 0.640·39-s + 1.98·43-s + 0.144·48-s − 1/7·49-s − 0.700·51-s − 0.554·52-s − 0.686·53-s + 1.40·61-s + 1/8·64-s − 0.606·68-s + 1.20·69-s + 0.461·75-s + 1.68·79-s + 1/9·81-s + 0.536·87-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: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 298116,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.149123677\)
\(L(\frac12)\) \(\approx\) \(2.149123677\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
13$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.5.a_ae
11$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.11.a_ad
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.17.f_u
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.23.ak_cs
29$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) 2.29.af_cg
31$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.31.a_bm
37$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \) 2.37.a_bj
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.a_as
43$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.43.an_es
47$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.47.a_ak
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.53.f_bo
59$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.59.a_an
61$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.al_ds
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.67.a_acs
71$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.71.a_adf
73$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.73.a_cn
79$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.79.ap_ia
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.83.a_afa
89$C_2^2$ \( 1 - 28 T^{2} + p^{2} T^{4} \) 2.89.a_abc
97$C_2^2$ \( 1 + 45 T^{2} + p^{2} T^{4} \) 2.97.a_bt
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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

−8.899198117292689499998569262421, −8.380847350327317158529368266934, −7.977331630548056092823738471662, −7.32771431900706671606297186236, −7.02369603426926566287949498718, −6.62382441085734171859436157986, −6.03347318609907285114506624894, −5.41741818021727729361528818109, −4.86184814183700460104797633128, −4.51976303116973255073994966655, −3.66336985164001797844791413720, −2.95796843295683148107206613295, −2.62449039200190061757987105243, −2.06833789359996254919216692370, −0.821908650580099209185103340300, 0.821908650580099209185103340300, 2.06833789359996254919216692370, 2.62449039200190061757987105243, 2.95796843295683148107206613295, 3.66336985164001797844791413720, 4.51976303116973255073994966655, 4.86184814183700460104797633128, 5.41741818021727729361528818109, 6.03347318609907285114506624894, 6.62382441085734171859436157986, 7.02369603426926566287949498718, 7.32771431900706671606297186236, 7.977331630548056092823738471662, 8.380847350327317158529368266934, 8.899198117292689499998569262421

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