Properties

Label 4-3850e2-1.1-c1e2-0-29
Degree $4$
Conductor $14822500$
Sign $1$
Analytic cond. $945.095$
Root an. cond. $5.54458$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 2·9-s + 2·11-s + 16-s − 8·19-s − 4·29-s − 20·31-s − 2·36-s − 2·44-s − 49-s − 20·59-s − 16·61-s − 64-s − 8·71-s + 8·76-s − 32·79-s − 5·81-s − 20·89-s + 4·99-s + 24·101-s + 28·109-s + 4·116-s + 3·121-s + 20·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1/2·4-s + 2/3·9-s + 0.603·11-s + 1/4·16-s − 1.83·19-s − 0.742·29-s − 3.59·31-s − 1/3·36-s − 0.301·44-s − 1/7·49-s − 2.60·59-s − 2.04·61-s − 1/8·64-s − 0.949·71-s + 0.917·76-s − 3.60·79-s − 5/9·81-s − 2.11·89-s + 0.402·99-s + 2.38·101-s + 2.68·109-s + 0.371·116-s + 3/11·121-s + 1.79·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14822500\)    =    \(2^{2} \cdot 5^{4} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(945.095\)
Root analytic conductor: \(5.54458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 14822500,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 158 T^{2} + p^{2} T^{4} \)
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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

−8.509142714804071880169470379014, −7.68097621618888915672921080039, −7.62612423498325200928407991979, −7.30365362569966069266190589503, −6.91231864705942975771342471721, −6.42637692479441654839206374861, −5.92437302492834489911484717312, −5.86126551682800339783548104855, −5.35915877197077491957819450341, −4.70665090138765286883566571405, −4.48509324650060004897813511000, −4.11690814864027198690782704995, −3.77622558572754632179191100898, −3.24280447332641273002125455754, −2.87957907785677397850662379278, −1.93571575960425887653193828978, −1.77598099464084405532041865821, −1.33820896313636744418261307428, 0, 0, 1.33820896313636744418261307428, 1.77598099464084405532041865821, 1.93571575960425887653193828978, 2.87957907785677397850662379278, 3.24280447332641273002125455754, 3.77622558572754632179191100898, 4.11690814864027198690782704995, 4.48509324650060004897813511000, 4.70665090138765286883566571405, 5.35915877197077491957819450341, 5.86126551682800339783548104855, 5.92437302492834489911484717312, 6.42637692479441654839206374861, 6.91231864705942975771342471721, 7.30365362569966069266190589503, 7.62612423498325200928407991979, 7.68097621618888915672921080039, 8.509142714804071880169470379014

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