Properties

Label 4-8624e2-1.1-c1e2-0-11
Degree $4$
Conductor $74373376$
Sign $1$
Analytic cond. $4742.11$
Root an. cond. $8.29837$
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  + 2·5-s + 9-s − 2·11-s + 10·13-s + 12·17-s + 6·19-s + 2·23-s + 2·29-s + 8·31-s + 2·37-s + 6·41-s + 8·43-s + 2·45-s − 16·47-s − 2·53-s − 4·55-s − 4·59-s + 18·61-s + 20·65-s + 8·67-s − 14·71-s + 6·73-s − 8·81-s − 16·83-s + 24·85-s − 8·89-s + 12·95-s + ⋯
L(s)  = 1  + 0.894·5-s + 1/3·9-s − 0.603·11-s + 2.77·13-s + 2.91·17-s + 1.37·19-s + 0.417·23-s + 0.371·29-s + 1.43·31-s + 0.328·37-s + 0.937·41-s + 1.21·43-s + 0.298·45-s − 2.33·47-s − 0.274·53-s − 0.539·55-s − 0.520·59-s + 2.30·61-s + 2.48·65-s + 0.977·67-s − 1.66·71-s + 0.702·73-s − 8/9·81-s − 1.75·83-s + 2.60·85-s − 0.847·89-s + 1.23·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74373376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74373376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(74373376\)    =    \(2^{8} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(4742.11\)
Root analytic conductor: \(8.29837\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 74373376,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.575287268\)
\(L(\frac12)\) \(\approx\) \(7.575287268\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
5$D_{4}$ \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 2 T + 40 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 2 T + 31 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 2 T + 68 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 16 T + 130 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T + 100 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 4 T + 115 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 18 T + 175 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 8 T + 87 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 14 T + 184 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 151 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 16 T + 202 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 22 T + 287 T^{2} + 22 p T^{3} + p^{2} T^{4} \)
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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

−7.977690224197909041124311325423, −7.904806215933294428776751238964, −7.16463159935505704374122427892, −6.96247438796743979447106023372, −6.39219917329537215989095654990, −6.25589967543361127439518887901, −5.70612710454375339430707773263, −5.69003880685543450766270264325, −5.20638565994174340742188263959, −5.13104302106102956406358208994, −4.28923402161824780917008581609, −4.01105118044731534360644395993, −3.68798666995465885207399270648, −3.16732717486347104028602724882, −2.80003698277562538141311438473, −2.77584002743352011984606403540, −1.71714779395425520239493176317, −1.31533137676839461678616572811, −1.18440293302880180007687105505, −0.73560815900512632775081729613, 0.73560815900512632775081729613, 1.18440293302880180007687105505, 1.31533137676839461678616572811, 1.71714779395425520239493176317, 2.77584002743352011984606403540, 2.80003698277562538141311438473, 3.16732717486347104028602724882, 3.68798666995465885207399270648, 4.01105118044731534360644395993, 4.28923402161824780917008581609, 5.13104302106102956406358208994, 5.20638565994174340742188263959, 5.69003880685543450766270264325, 5.70612710454375339430707773263, 6.25589967543361127439518887901, 6.39219917329537215989095654990, 6.96247438796743979447106023372, 7.16463159935505704374122427892, 7.904806215933294428776751238964, 7.977690224197909041124311325423

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