Properties

Label 4-2450e2-1.1-c1e2-0-29
Degree $4$
Conductor $6002500$
Sign $1$
Analytic cond. $382.724$
Root an. cond. $4.42304$
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  + 2·2-s + 3·4-s + 4·8-s − 6·9-s − 8·11-s + 5·16-s − 12·18-s − 16·22-s − 8·29-s + 6·32-s − 18·36-s − 12·37-s − 24·43-s − 24·44-s − 24·53-s − 16·58-s + 7·64-s − 24·67-s + 16·71-s − 24·72-s − 24·74-s + 27·81-s − 48·86-s − 32·88-s + 48·99-s − 48·106-s + 24·107-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s − 2·9-s − 2.41·11-s + 5/4·16-s − 2.82·18-s − 3.41·22-s − 1.48·29-s + 1.06·32-s − 3·36-s − 1.97·37-s − 3.65·43-s − 3.61·44-s − 3.29·53-s − 2.10·58-s + 7/8·64-s − 2.93·67-s + 1.89·71-s − 2.82·72-s − 2.78·74-s + 3·81-s − 5.17·86-s − 3.41·88-s + 4.82·99-s − 4.66·106-s + 2.32·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6002500\)    =    \(2^{2} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(382.724\)
Root analytic conductor: \(4.42304\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2450} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 6002500,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
7 \( 1 \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 72 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 128 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 122 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 160 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 176 T^{2} + p^{2} T^{4} \)
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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

−8.404936149144031677513775353019, −8.363890546288383534446644373507, −7.951527539037869539182624015648, −7.64676227678662275011660042611, −7.13864555412994918477703332759, −6.70412314992962757671132540921, −6.21616518601793966339405085165, −5.93237179282087313858039130507, −5.37306626198236309732120880153, −5.28414901579079646322877809444, −4.89804885205522541521761378318, −4.63838188154805417797408088974, −3.64122797614887584500242126553, −3.39007701798454149535875122819, −2.97785655923287429873323717644, −2.79079397579378826809843370943, −1.88171923399790662800702456563, −1.84006062092222343779552731877, 0, 0, 1.84006062092222343779552731877, 1.88171923399790662800702456563, 2.79079397579378826809843370943, 2.97785655923287429873323717644, 3.39007701798454149535875122819, 3.64122797614887584500242126553, 4.63838188154805417797408088974, 4.89804885205522541521761378318, 5.28414901579079646322877809444, 5.37306626198236309732120880153, 5.93237179282087313858039130507, 6.21616518601793966339405085165, 6.70412314992962757671132540921, 7.13864555412994918477703332759, 7.64676227678662275011660042611, 7.951527539037869539182624015648, 8.363890546288383534446644373507, 8.404936149144031677513775353019

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