Properties

Label 8-1216e4-1.1-c1e4-0-14
Degree $8$
Conductor $2.186\times 10^{12}$
Sign $1$
Analytic cond. $8888.79$
Root an. cond. $3.11605$
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  + 12·9-s + 28·17-s − 18·25-s + 10·49-s − 44·73-s + 90·81-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 336·153-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 4·9-s + 6.79·17-s − 3.59·25-s + 10/7·49-s − 5.14·73-s + 10·81-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 27.1·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(8888.79\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(9.129627149\)
\(L(\frac12)\) \(\approx\) \(9.129627149\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( ( 1 - p T^{2} )^{2} \)
good3$C_2$ \( ( 1 - p T^{2} )^{4} \)
5$C_2$ \( ( 1 - T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \)
7$C_2^2$ \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 3 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 + p T^{2} )^{4} \)
17$C_2$ \( ( 1 - 7 T + p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2$ \( ( 1 + p T^{2} )^{4} \)
41$C_2$ \( ( 1 - p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 75 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{4} \)
59$C_2$ \( ( 1 - p T^{2} )^{4} \)
61$C_2$ \( ( 1 - 15 T + p T^{2} )^{2}( 1 + 15 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - p T^{2} )^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 90 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 - p T^{2} )^{4} \)
97$C_2$ \( ( 1 - p T^{2} )^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.28019754241498009659040552412, −6.88805845467818410222614558906, −6.61376572303761601579400519709, −6.03004530056885962766843121805, −5.98560820625299776971249702804, −5.81263620916502455403853978296, −5.78458890582269204569409955706, −5.43449457513465448777278228174, −5.23385125515505769292448144974, −4.96565149590601462395531688317, −4.47050555008448355340762828076, −4.46666430004873874130089099762, −4.23258953774493031647381405989, −3.81429869027980297976604024187, −3.73079242443933255429529381569, −3.52967760534338542679190086557, −3.36368190604318474366024854417, −2.97723176560102034372782779674, −2.72553942747422802383078857394, −1.93680931932205370632783871178, −1.89562383277427211350024887206, −1.52586310849451429400949758104, −1.28901180262361881692405061256, −1.06792972209379539850200108938, −0.69911046023946851255477402298, 0.69911046023946851255477402298, 1.06792972209379539850200108938, 1.28901180262361881692405061256, 1.52586310849451429400949758104, 1.89562383277427211350024887206, 1.93680931932205370632783871178, 2.72553942747422802383078857394, 2.97723176560102034372782779674, 3.36368190604318474366024854417, 3.52967760534338542679190086557, 3.73079242443933255429529381569, 3.81429869027980297976604024187, 4.23258953774493031647381405989, 4.46666430004873874130089099762, 4.47050555008448355340762828076, 4.96565149590601462395531688317, 5.23385125515505769292448144974, 5.43449457513465448777278228174, 5.78458890582269204569409955706, 5.81263620916502455403853978296, 5.98560820625299776971249702804, 6.03004530056885962766843121805, 6.61376572303761601579400519709, 6.88805845467818410222614558906, 7.28019754241498009659040552412

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