Properties

Label 4-204e2-1.1-c1e2-0-15
Degree $4$
Conductor $41616$
Sign $-1$
Analytic cond. $2.65347$
Root an. cond. $1.27630$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 3·8-s − 9-s − 4·13-s − 16-s − 6·17-s − 18-s − 8·19-s − 6·25-s − 4·26-s + 5·32-s − 6·34-s + 36-s − 8·38-s + 16·43-s + 8·47-s + 6·49-s − 6·50-s + 4·52-s − 12·53-s + 7·64-s − 8·67-s + 6·68-s + 3·72-s + 8·76-s + 81-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s − 1/3·9-s − 1.10·13-s − 1/4·16-s − 1.45·17-s − 0.235·18-s − 1.83·19-s − 6/5·25-s − 0.784·26-s + 0.883·32-s − 1.02·34-s + 1/6·36-s − 1.29·38-s + 2.43·43-s + 1.16·47-s + 6/7·49-s − 0.848·50-s + 0.554·52-s − 1.64·53-s + 7/8·64-s − 0.977·67-s + 0.727·68-s + 0.353·72-s + 0.917·76-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(41616\)    =    \(2^{4} \cdot 3^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(2.65347\)
Root analytic conductor: \(1.27630\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{41616} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 41616,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - T + p T^{2} \)
3$C_2$ \( 1 + T^{2} \)
17$C_2$ \( 1 + 6 T + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
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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

−10.04482375723634155775764619731, −9.256284747915140423961708364403, −9.042029466165061074979809595601, −8.597242461430919083768986613318, −7.81217159928536431677143053445, −7.41364142528798531159935494679, −6.48745340244325304717557907543, −6.18674731500408210746746546183, −5.54781665209094954178002207182, −4.83204129687325674921674222381, −4.23507525077229997330316946259, −3.96021101701915957778218066594, −2.71548516817632809510348884503, −2.22806382313505085281828995013, 0, 2.22806382313505085281828995013, 2.71548516817632809510348884503, 3.96021101701915957778218066594, 4.23507525077229997330316946259, 4.83204129687325674921674222381, 5.54781665209094954178002207182, 6.18674731500408210746746546183, 6.48745340244325304717557907543, 7.41364142528798531159935494679, 7.81217159928536431677143053445, 8.597242461430919083768986613318, 9.042029466165061074979809595601, 9.256284747915140423961708364403, 10.04482375723634155775764619731

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