Properties

Label 4-5070e2-1.1-c1e2-0-23
Degree $4$
Conductor $25704900$
Sign $1$
Analytic cond. $1638.96$
Root an. cond. $6.36271$
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  + 2·3-s − 4-s + 3·9-s − 2·12-s + 16-s − 12·17-s − 25-s + 4·27-s − 12·29-s − 3·36-s + 8·43-s + 2·48-s − 2·49-s − 24·51-s − 12·53-s − 20·61-s − 64-s + 12·68-s − 2·75-s + 16·79-s + 5·81-s − 24·87-s + 100-s − 36·101-s + 8·103-s − 24·107-s − 4·108-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s + 9-s − 0.577·12-s + 1/4·16-s − 2.91·17-s − 1/5·25-s + 0.769·27-s − 2.22·29-s − 1/2·36-s + 1.21·43-s + 0.288·48-s − 2/7·49-s − 3.36·51-s − 1.64·53-s − 2.56·61-s − 1/8·64-s + 1.45·68-s − 0.230·75-s + 1.80·79-s + 5/9·81-s − 2.57·87-s + 1/10·100-s − 3.58·101-s + 0.788·103-s − 2.32·107-s − 0.384·108-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(25704900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(1638.96\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5070} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 25704900,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + T^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
5$C_2$ \( 1 + T^{2} \)
13 \( 1 \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 146 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−8.127556002279082142514529988207, −7.73562877633696931316770037680, −7.38060663391904745492757492819, −7.16198049135809868549882999993, −6.58050436458872900406455748756, −6.30364501080472299132614678930, −6.06106737387786767843239252016, −5.39791813652783019594829251342, −4.87711270690896588078666881478, −4.81585939898988208509292805131, −4.08369078714151938323789688344, −3.97252259699027145899570373537, −3.71790943787385589931687958200, −2.96959076936302955068544631647, −2.54550686686188345017479747572, −2.33400265261670445054979910362, −1.60991187752735837712633891342, −1.40311373771204694994124485920, 0, 0, 1.40311373771204694994124485920, 1.60991187752735837712633891342, 2.33400265261670445054979910362, 2.54550686686188345017479747572, 2.96959076936302955068544631647, 3.71790943787385589931687958200, 3.97252259699027145899570373537, 4.08369078714151938323789688344, 4.81585939898988208509292805131, 4.87711270690896588078666881478, 5.39791813652783019594829251342, 6.06106737387786767843239252016, 6.30364501080472299132614678930, 6.58050436458872900406455748756, 7.16198049135809868549882999993, 7.38060663391904745492757492819, 7.73562877633696931316770037680, 8.127556002279082142514529988207

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