Properties

Label 4-2850e2-1.1-c1e2-0-4
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 9-s + 16-s − 2·19-s − 4·29-s − 16·31-s + 36-s + 12·41-s − 2·49-s + 24·59-s − 4·61-s − 64-s − 16·71-s + 2·76-s + 32·79-s + 81-s − 12·89-s − 36·101-s − 12·109-s + 4·116-s − 22·121-s + 16·124-s + 127-s + 131-s + 137-s + 139-s − 144-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s + 1/4·16-s − 0.458·19-s − 0.742·29-s − 2.87·31-s + 1/6·36-s + 1.87·41-s − 2/7·49-s + 3.12·59-s − 0.512·61-s − 1/8·64-s − 1.89·71-s + 0.229·76-s + 3.60·79-s + 1/9·81-s − 1.27·89-s − 3.58·101-s − 1.14·109-s + 0.371·116-s − 2·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-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: \(0\)
Selberg data: \((4,\ 8122500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9353998347\)
\(L(\frac12)\) \(\approx\) \(0.9353998347\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
19$C_1$ \( ( 1 + T )^{2} \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−9.185434879203501535050741887237, −8.618374181995704999307424050565, −8.153685694424773513054847193544, −7.988183746019224980298770398170, −7.40976707091801464701384535056, −7.16643917689858276515281674832, −6.75990377403134833462161881900, −6.29011011853380053769828032362, −5.73685866075118849776137662862, −5.50364735398736847891208494219, −5.26851736532864244408542620485, −4.68713193681521560981260189493, −4.03559650739469845883376899103, −3.93355982653213919556634538434, −3.51319819809025100368643252288, −2.81561454243333712503913721757, −2.36823222183087952222672230080, −1.84806203903419605008932962558, −1.19556409811313956677762926692, −0.32847670179714550338281588766, 0.32847670179714550338281588766, 1.19556409811313956677762926692, 1.84806203903419605008932962558, 2.36823222183087952222672230080, 2.81561454243333712503913721757, 3.51319819809025100368643252288, 3.93355982653213919556634538434, 4.03559650739469845883376899103, 4.68713193681521560981260189493, 5.26851736532864244408542620485, 5.50364735398736847891208494219, 5.73685866075118849776137662862, 6.29011011853380053769828032362, 6.75990377403134833462161881900, 7.16643917689858276515281674832, 7.40976707091801464701384535056, 7.988183746019224980298770398170, 8.153685694424773513054847193544, 8.618374181995704999307424050565, 9.185434879203501535050741887237

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