Properties

Label 4-2850e2-1.1-c1e2-0-24
Degree $4$
Conductor $8122500$
Sign $1$
Analytic cond. $517.897$
Root an. cond. $4.77046$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 3·4-s + 4·6-s − 2·7-s − 4·8-s + 3·9-s + 4·11-s − 6·12-s − 2·13-s + 4·14-s + 5·16-s − 2·17-s − 6·18-s − 2·19-s + 4·21-s − 8·22-s − 2·23-s + 8·24-s + 4·26-s − 4·27-s − 6·28-s + 2·31-s − 6·32-s − 8·33-s + 4·34-s + 9·36-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s + 1.20·11-s − 1.73·12-s − 0.554·13-s + 1.06·14-s + 5/4·16-s − 0.485·17-s − 1.41·18-s − 0.458·19-s + 0.872·21-s − 1.70·22-s − 0.417·23-s + 1.63·24-s + 0.784·26-s − 0.769·27-s − 1.13·28-s + 0.359·31-s − 1.06·32-s − 1.39·33-s + 0.685·34-s + 3/2·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8122500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(517.897\)
Root analytic conductor: \(4.77046\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2850} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 8122500,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
19$C_1$ \( ( 1 + T )^{2} \)
good7$D_{4}$ \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 4 T + 23 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 2 T + 51 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 6 T + 68 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 103 T^{2} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T + 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 135 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 14 T + 164 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 2 T + 135 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T + 167 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 6 T + 56 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
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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

−8.589637006218528387421201326961, −8.360220202259788747413491715663, −7.70982633546656140732920970351, −7.59551879178427233119901295986, −6.86463295501935493794246110714, −6.76204246059534951640029166572, −6.50337156095344798948573195491, −6.12746734437435725474151843130, −5.69493695338968508555443379806, −5.28864338973732717151319425623, −4.60046546227079517422315404741, −4.41123081895737540513744296132, −3.58457526565963067757022467920, −3.50897135866188363673771118030, −2.51491091965069771385375490574, −2.31132162016088562257512473367, −1.36647173771941590272376626945, −1.22005894131790457823118301979, 0, 0, 1.22005894131790457823118301979, 1.36647173771941590272376626945, 2.31132162016088562257512473367, 2.51491091965069771385375490574, 3.50897135866188363673771118030, 3.58457526565963067757022467920, 4.41123081895737540513744296132, 4.60046546227079517422315404741, 5.28864338973732717151319425623, 5.69493695338968508555443379806, 6.12746734437435725474151843130, 6.50337156095344798948573195491, 6.76204246059534951640029166572, 6.86463295501935493794246110714, 7.59551879178427233119901295986, 7.70982633546656140732920970351, 8.360220202259788747413491715663, 8.589637006218528387421201326961

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