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Properties

Label 4-290e2-1.1-c1e2-0-23
Degree $4$
Conductor $84100$
Sign $-1$
Analytic cond. $5.36228$
Root an. cond. $1.52172$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 5-s − 9-s − 6·11-s + 16-s − 20-s − 4·25-s + 2·29-s − 14·31-s − 36-s + 8·41-s − 6·44-s + 45-s − 2·49-s + 6·55-s − 4·59-s + 4·61-s + 64-s − 20·71-s − 22·79-s − 80-s − 8·81-s + 6·99-s − 4·100-s + 16·101-s + 2·109-s + 2·116-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.447·5-s − 1/3·9-s − 1.80·11-s + 1/4·16-s − 0.223·20-s − 4/5·25-s + 0.371·29-s − 2.51·31-s − 1/6·36-s + 1.24·41-s − 0.904·44-s + 0.149·45-s − 2/7·49-s + 0.809·55-s − 0.520·59-s + 0.512·61-s + 1/8·64-s − 2.37·71-s − 2.47·79-s − 0.111·80-s − 8/9·81-s + 0.603·99-s − 2/5·100-s + 1.59·101-s + 0.191·109-s + 0.185·116-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(84100\)    =    \(2^{2} \cdot 5^{2} \cdot 29^{2}\)
Sign: $-1$
Analytic conductor: \(5.36228\)
Root analytic conductor: \(1.52172\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 84100,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_2$ \( 1 + T + p T^{2} \)
29$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.11.g_bb
13$C_2^2$ \( 1 - 15 T^{2} + p^{2} T^{4} \) 2.13.a_ap
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.17.a_k
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.23.a_abi
31$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.31.o_ed
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.a_bm
41$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.41.ai_bi
43$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \) 2.43.a_ax
47$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \) 2.47.a_ah
53$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \) 2.53.a_r
59$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.59.e_aw
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.ae_ck
67$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.67.a_cw
71$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.71.u_is
73$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.73.a_be
79$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.79.w_jj
83$C_2^2$ \( 1 + 134 T^{2} + p^{2} T^{4} \) 2.83.a_fe
89$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.89.a_ek
97$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.97.a_ahm
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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

−9.484633320833228349581498017186, −8.839735193779006045394842987948, −8.462577963295780227342794972187, −7.76643223735093624812883349253, −7.46895723629394649810813192504, −7.19939270998070999334336100694, −6.26742052107305117261919960732, −5.67797853663145081086873545770, −5.44208114512005954876917038247, −4.62547299646254388948411905764, −3.96819737653166668580252003758, −3.15829880853553865088623924262, −2.62661510122108731021638197128, −1.77348092460854087171065116547, 0, 1.77348092460854087171065116547, 2.62661510122108731021638197128, 3.15829880853553865088623924262, 3.96819737653166668580252003758, 4.62547299646254388948411905764, 5.44208114512005954876917038247, 5.67797853663145081086873545770, 6.26742052107305117261919960732, 7.19939270998070999334336100694, 7.46895723629394649810813192504, 7.76643223735093624812883349253, 8.462577963295780227342794972187, 8.839735193779006045394842987948, 9.484633320833228349581498017186

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