Properties

Label 4-442e2-1.1-c1e2-0-8
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 − 2·9-s − 2·13-s + 5·16-s + 6·17-s + 4·18-s + 12·19-s + 10·25-s + 4·26-s − 6·32-s − 12·34-s − 6·36-s − 24·38-s − 4·43-s − 12·47-s − 2·49-s − 20·50-s − 6·52-s + 12·53-s + 7·64-s + 18·68-s + 8·72-s + 36·76-s − 5·81-s − 12·83-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s − 2/3·9-s − 0.554·13-s + 5/4·16-s + 1.45·17-s + 0.942·18-s + 2.75·19-s + 2·25-s + 0.784·26-s − 1.06·32-s − 2.05·34-s − 36-s − 3.89·38-s − 0.609·43-s − 1.75·47-s − 2/7·49-s − 2.82·50-s − 0.832·52-s + 1.64·53-s + 7/8·64-s + 2.18·68-s + 0.942·72-s + 4.12·76-s − 5/9·81-s − 1.31·83-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

\(L(1)\) \(\approx\) \(0.9018215483\)
\(L(\frac12)\) \(\approx\) \(0.9018215483\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 2 T + p T^{2} \)
17$C_2$ \( 1 - 6 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
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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

−9.004623219661312748024203346663, −8.766816561979011140954392317228, −8.144435992888895960206435617566, −7.77751673161650612750438794458, −7.28141405597615369212003587151, −6.99861974655900237767063112815, −6.35876926744938798864350964901, −5.63736860850390702119072246291, −5.28600360410984827458958153613, −4.82938658159563541189113513207, −3.55893118767353223721678546123, −3.09428708935862751937248857990, −2.70868561347981370113654207898, −1.50654215239458831044175694911, −0.826144107334518372930392995703, 0.826144107334518372930392995703, 1.50654215239458831044175694911, 2.70868561347981370113654207898, 3.09428708935862751937248857990, 3.55893118767353223721678546123, 4.82938658159563541189113513207, 5.28600360410984827458958153613, 5.63736860850390702119072246291, 6.35876926744938798864350964901, 6.99861974655900237767063112815, 7.28141405597615369212003587151, 7.77751673161650612750438794458, 8.144435992888895960206435617566, 8.766816561979011140954392317228, 9.004623219661312748024203346663

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