Properties

Label 4-442e2-1.1-c1e2-0-29
Degree $4$
Conductor $195364$
Sign $1$
Analytic cond. $12.4565$
Root an. cond. $1.87866$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s + 4·8-s + 4·9-s + 4·13-s + 5·16-s + 8·18-s − 6·19-s − 2·25-s + 8·26-s + 6·32-s + 12·36-s − 12·38-s + 2·43-s − 12·47-s + 10·49-s − 4·50-s + 12·52-s + 18·53-s − 24·59-s + 7·64-s + 12·67-s + 16·72-s − 18·76-s + 7·81-s − 6·83-s + 4·86-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s + 4/3·9-s + 1.10·13-s + 5/4·16-s + 1.88·18-s − 1.37·19-s − 2/5·25-s + 1.56·26-s + 1.06·32-s + 2·36-s − 1.94·38-s + 0.304·43-s − 1.75·47-s + 10/7·49-s − 0.565·50-s + 1.66·52-s + 2.47·53-s − 3.12·59-s + 7/8·64-s + 1.46·67-s + 1.88·72-s − 2.06·76-s + 7/9·81-s − 0.658·83-s + 0.431·86-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(195364\)    =    \(2^{2} \cdot 13^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(12.4565\)
Root analytic conductor: \(1.87866\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{195364} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 195364,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.788265656\)
\(L(\frac12)\) \(\approx\) \(4.788265656\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
13$C_2$ \( 1 - 4 T + p T^{2} \)
17$C_2$ \( 1 + p T^{2} \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
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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

−9.013380651274237440960027820503, −8.630795411634816744144466492179, −7.957665828749720952244238653045, −7.58890477284572747881969749852, −6.89954683716125494045405132044, −6.66996008656293390414235256590, −6.07145866022164271478248397461, −5.71563398774331333479442976650, −4.97039917917502874590256560627, −4.47424351973555139425754812103, −3.95930014054124244019054092449, −3.69217744196914456938365884781, −2.78317735857444085757743491047, −2.02399240856807550643595824193, −1.31573896657370614619554593443, 1.31573896657370614619554593443, 2.02399240856807550643595824193, 2.78317735857444085757743491047, 3.69217744196914456938365884781, 3.95930014054124244019054092449, 4.47424351973555139425754812103, 4.97039917917502874590256560627, 5.71563398774331333479442976650, 6.07145866022164271478248397461, 6.66996008656293390414235256590, 6.89954683716125494045405132044, 7.58890477284572747881969749852, 7.957665828749720952244238653045, 8.630795411634816744144466492179, 9.013380651274237440960027820503

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