Properties

Label 4-2850e2-1.1-c1e2-0-1
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 − 12·11-s + 16-s + 2·19-s + 16·29-s − 16·31-s + 36-s − 8·41-s + 12·44-s + 10·49-s + 8·59-s + 4·61-s − 64-s − 2·76-s − 16·79-s + 81-s + 8·89-s + 12·99-s − 20·101-s + 28·109-s − 16·116-s + 86·121-s + 16·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s − 3.61·11-s + 1/4·16-s + 0.458·19-s + 2.97·29-s − 2.87·31-s + 1/6·36-s − 1.24·41-s + 1.80·44-s + 10/7·49-s + 1.04·59-s + 0.512·61-s − 1/8·64-s − 0.229·76-s − 1.80·79-s + 1/9·81-s + 0.847·89-s + 1.20·99-s − 1.99·101-s + 2.68·109-s − 1.48·116-s + 7.81·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-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

\(L(1)\) \(\approx\) \(0.4532424361\)
\(L(\frac12)\) \(\approx\) \(0.4532424361\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
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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

−8.748405958471388080676542011185, −8.514363960329833016188459269758, −8.195120690690702149987255975855, −8.070432006248144779169048415832, −7.37365270402183415445398658467, −7.10384103306805227964925950604, −7.07434882209134449263965328751, −5.95815623256053133809195794119, −5.92986547221285387275910221698, −5.30420021914845587654280960194, −5.26721247081828854607955720388, −4.67999193834739027564263963275, −4.56862626690688092125252795506, −3.57599299436849554045616775686, −3.40033153489839189689044095054, −2.75102892443070551879861934766, −2.48236430737732112372111426490, −2.03729841766325524846121265926, −1.03826223807395454268976341650, −0.24166529714376774706309971269, 0.24166529714376774706309971269, 1.03826223807395454268976341650, 2.03729841766325524846121265926, 2.48236430737732112372111426490, 2.75102892443070551879861934766, 3.40033153489839189689044095054, 3.57599299436849554045616775686, 4.56862626690688092125252795506, 4.67999193834739027564263963275, 5.26721247081828854607955720388, 5.30420021914845587654280960194, 5.92986547221285387275910221698, 5.95815623256053133809195794119, 7.07434882209134449263965328751, 7.10384103306805227964925950604, 7.37365270402183415445398658467, 8.070432006248144779169048415832, 8.195120690690702149987255975855, 8.514363960329833016188459269758, 8.748405958471388080676542011185

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