Properties

Label 4-3327500-1.1-c1e2-0-13
Degree $4$
Conductor $3327500$
Sign $-1$
Analytic cond. $212.164$
Root an. cond. $3.81652$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4-s − 3·9-s + 11-s + 2·12-s + 16-s + 12·23-s − 14·27-s − 6·31-s + 2·33-s − 3·36-s − 6·37-s + 44-s + 4·47-s + 2·48-s − 5·49-s + 2·53-s − 20·59-s + 64-s − 16·67-s + 24·69-s + 14·71-s − 4·81-s − 30·89-s + 12·92-s − 12·93-s + 24·97-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/2·4-s − 9-s + 0.301·11-s + 0.577·12-s + 1/4·16-s + 2.50·23-s − 2.69·27-s − 1.07·31-s + 0.348·33-s − 1/2·36-s − 0.986·37-s + 0.150·44-s + 0.583·47-s + 0.288·48-s − 5/7·49-s + 0.274·53-s − 2.60·59-s + 1/8·64-s − 1.95·67-s + 2.88·69-s + 1.66·71-s − 4/9·81-s − 3.17·89-s + 1.25·92-s − 1.24·93-s + 2.43·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3327500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3327500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3327500\)    =    \(2^{2} \cdot 5^{4} \cdot 11^{3}\)
Sign: $-1$
Analytic conductor: \(212.164\)
Root analytic conductor: \(3.81652\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 3327500,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
11$C_1$ \( 1 - T \)
good3$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.3.ac_h
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.17.a_ap
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.19.a_n
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.23.am_de
29$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.29.a_bh
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.31.g_ct
37$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.37.g_df
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.a_da
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.47.ae_du
53$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.53.ac_ed
59$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.59.u_ik
61$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.61.a_cv
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.67.q_hq
71$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.71.ao_hj
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.a_aby
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.a_cg
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \) 2.89.be_pn
97$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.97.ay_na
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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

−7.35838560113530362823348866724, −7.15522824999053493291691011446, −6.32452856749704006618306134382, −6.25653525333154357749893379339, −5.51294812234663797155432190910, −5.35665492802824494420800719571, −4.74461890335682791263484236396, −4.24248350881714595932270404880, −3.46996281594193181675719351221, −3.13633289039528983743583004611, −3.07694177217990984214438626883, −2.32256149854148039372877464389, −1.86665970315543630585959813737, −1.14442070453994046878747912184, 0, 1.14442070453994046878747912184, 1.86665970315543630585959813737, 2.32256149854148039372877464389, 3.07694177217990984214438626883, 3.13633289039528983743583004611, 3.46996281594193181675719351221, 4.24248350881714595932270404880, 4.74461890335682791263484236396, 5.35665492802824494420800719571, 5.51294812234663797155432190910, 6.25653525333154357749893379339, 6.32452856749704006618306134382, 7.15522824999053493291691011446, 7.35838560113530362823348866724

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