Properties

Label 4-550e2-1.1-c1e2-0-1
Degree $4$
Conductor $302500$
Sign $1$
Analytic cond. $19.2876$
Root an. cond. $2.09565$
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  − 4-s − 5·9-s + 3·11-s + 16-s − 4·31-s + 5·36-s − 3·44-s + 10·49-s − 64-s + 24·71-s + 16·81-s − 30·89-s − 15·99-s − 2·121-s + 4·124-s + 127-s + 131-s + 137-s + 139-s − 5·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + ⋯
L(s)  = 1  − 1/2·4-s − 5/3·9-s + 0.904·11-s + 1/4·16-s − 0.718·31-s + 5/6·36-s − 0.452·44-s + 10/7·49-s − 1/8·64-s + 2.84·71-s + 16/9·81-s − 3.17·89-s − 1.50·99-s − 0.181·121-s + 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.416·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 302500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 302500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(302500\)    =    \(2^{2} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(19.2876\)
Root analytic conductor: \(2.09565\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 302500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.152207629\)
\(L(\frac12)\) \(\approx\) \(1.152207629\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
11$C_2$ \( 1 - 3 T + p T^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−8.762789345040393770618893038027, −8.551063144214061160314341107343, −7.957724322686591184846060187353, −7.55692819541380612335203925812, −6.79746415632834177316334213207, −6.52149036696622551353959615452, −5.84966477610753839770841694590, −5.43345585786008524862229800874, −5.13439819891690129641175915714, −4.19197004093773331929479716587, −3.91634035105549994628398222605, −3.17725005272790474159542129978, −2.64302805183391542069297874498, −1.78541078077555676954287571857, −0.62943434565369051457038918567, 0.62943434565369051457038918567, 1.78541078077555676954287571857, 2.64302805183391542069297874498, 3.17725005272790474159542129978, 3.91634035105549994628398222605, 4.19197004093773331929479716587, 5.13439819891690129641175915714, 5.43345585786008524862229800874, 5.84966477610753839770841694590, 6.52149036696622551353959615452, 6.79746415632834177316334213207, 7.55692819541380612335203925812, 7.957724322686591184846060187353, 8.551063144214061160314341107343, 8.762789345040393770618893038027

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