Properties

Label 4-758912-1.1-c1e2-0-65
Degree $4$
Conductor $758912$
Sign $-1$
Analytic cond. $48.3888$
Root an. cond. $2.63746$
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 + 8-s + 2·9-s − 2·11-s + 16-s + 2·18-s − 2·22-s − 6·25-s + 32-s + 2·36-s − 24·43-s − 2·44-s − 7·49-s − 6·50-s + 64-s − 24·67-s + 2·72-s − 5·81-s − 24·86-s − 2·88-s − 7·98-s − 4·99-s − 6·100-s + 24·107-s + 20·113-s + 3·121-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s + 2/3·9-s − 0.603·11-s + 1/4·16-s + 0.471·18-s − 0.426·22-s − 6/5·25-s + 0.176·32-s + 1/3·36-s − 3.65·43-s − 0.301·44-s − 49-s − 0.848·50-s + 1/8·64-s − 2.93·67-s + 0.235·72-s − 5/9·81-s − 2.58·86-s − 0.213·88-s − 0.707·98-s − 0.402·99-s − 3/5·100-s + 2.32·107-s + 1.88·113-s + 3/11·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 758912 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 758912 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(758912\)    =    \(2^{7} \cdot 7^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(48.3888\)
Root analytic conductor: \(2.63746\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 758912,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
7$C_2$ \( 1 + p T^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 158 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
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

−7.956688321669486949443993345122, −7.44503492091689427214437925251, −7.28384383496805176593505108744, −6.58795954861217297502080464861, −6.22829773044480341331555491835, −5.83925184008878762369803223001, −5.11224655744256040433863219406, −4.88550380629058443410670715554, −4.39507406197561839253841275171, −3.72425550395436005402207348894, −3.31879301455778368319659817430, −2.74561130040773494403748773066, −1.89204299677024356271142659476, −1.50252589523985056842616413160, 0, 1.50252589523985056842616413160, 1.89204299677024356271142659476, 2.74561130040773494403748773066, 3.31879301455778368319659817430, 3.72425550395436005402207348894, 4.39507406197561839253841275171, 4.88550380629058443410670715554, 5.11224655744256040433863219406, 5.83925184008878762369803223001, 6.22829773044480341331555491835, 6.58795954861217297502080464861, 7.28384383496805176593505108744, 7.44503492091689427214437925251, 7.956688321669486949443993345122

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