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Properties

Label 4-220e2-1.1-c1e2-0-4
Degree $4$
Conductor $48400$
Sign $-1$
Analytic cond. $3.08602$
Root an. cond. $1.32540$
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  − 4·3-s − 2·5-s + 6·9-s + 8·15-s + 12·23-s + 3·25-s + 4·27-s − 8·31-s + 4·37-s − 12·45-s − 12·47-s − 10·49-s − 12·53-s + 24·59-s + 4·67-s − 48·69-s − 24·71-s − 12·75-s − 37·81-s − 12·89-s + 32·93-s + 4·97-s + 28·103-s − 16·111-s − 12·113-s − 24·115-s − 11·121-s + ⋯
L(s)  = 1  − 2.30·3-s − 0.894·5-s + 2·9-s + 2.06·15-s + 2.50·23-s + 3/5·25-s + 0.769·27-s − 1.43·31-s + 0.657·37-s − 1.78·45-s − 1.75·47-s − 1.42·49-s − 1.64·53-s + 3.12·59-s + 0.488·67-s − 5.77·69-s − 2.84·71-s − 1.38·75-s − 4.11·81-s − 1.27·89-s + 3.31·93-s + 0.406·97-s + 2.75·103-s − 1.51·111-s − 1.12·113-s − 2.23·115-s − 121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(48400\)    =    \(2^{4} \cdot 5^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(3.08602\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 48400,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
11$C_2$ \( 1 + p T^{2} \)
good3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.3.e_k
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.23.am_de
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.a_ao
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.47.m_fa
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.a_eo
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.67.ae_fi
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.a_dq
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
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

−10.01348145264732284453142036013, −9.549091370160705845855623611865, −8.597075432672291247955228431981, −8.552217550781204179646223792184, −7.48339952967373098458922175110, −7.12025897518162428207793885558, −6.57891116465648258947670054106, −6.09088770920212572077792597538, −5.46752609716119299096669229449, −4.83331995302090139893539895632, −4.78130792717525308450176413839, −3.61120105048083329633068803983, −2.90312545354050643736888796120, −1.17768193714929585417069498904, 0, 1.17768193714929585417069498904, 2.90312545354050643736888796120, 3.61120105048083329633068803983, 4.78130792717525308450176413839, 4.83331995302090139893539895632, 5.46752609716119299096669229449, 6.09088770920212572077792597538, 6.57891116465648258947670054106, 7.12025897518162428207793885558, 7.48339952967373098458922175110, 8.552217550781204179646223792184, 8.597075432672291247955228431981, 9.549091370160705845855623611865, 10.01348145264732284453142036013

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