Properties

Label 4-1229312-1.1-c1e2-0-38
Degree $4$
Conductor $1229312$
Sign $1$
Analytic cond. $78.3819$
Root an. cond. $2.97545$
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  − 6·9-s − 8·11-s + 12·17-s − 16·19-s − 6·25-s − 4·41-s − 8·43-s − 8·67-s − 20·73-s + 27·81-s − 16·83-s + 12·89-s + 12·97-s + 48·99-s − 24·107-s + 4·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 72·153-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 2·9-s − 2.41·11-s + 2.91·17-s − 3.67·19-s − 6/5·25-s − 0.624·41-s − 1.21·43-s − 0.977·67-s − 2.34·73-s + 3·81-s − 1.75·83-s + 1.27·89-s + 1.21·97-s + 4.82·99-s − 2.32·107-s + 0.376·113-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 5.82·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1229312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1229312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1229312\)    =    \(2^{9} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(78.3819\)
Root analytic conductor: \(2.97545\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1229312,\ (\ :1/2, 1/2),\ 1)\)

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

−7.84054312303988042972335609462, −7.30264187759595822059178656689, −6.59264336057149769206055975154, −5.92791093661634117809390012383, −5.91740132798549001211232285063, −5.47493819124631815597461836959, −4.98409920426632935260415692638, −4.53211800578457486655010876487, −3.68340574560716411313233602487, −3.32773603072291404011890923635, −2.58182243115007516792385368988, −2.50561452898491372410349134801, −1.62407385510972851799092712087, 0, 0, 1.62407385510972851799092712087, 2.50561452898491372410349134801, 2.58182243115007516792385368988, 3.32773603072291404011890923635, 3.68340574560716411313233602487, 4.53211800578457486655010876487, 4.98409920426632935260415692638, 5.47493819124631815597461836959, 5.91740132798549001211232285063, 5.92791093661634117809390012383, 6.59264336057149769206055975154, 7.30264187759595822059178656689, 7.84054312303988042972335609462

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