Properties

Label 4-1148e2-1.1-c1e2-0-10
Degree $4$
Conductor $1317904$
Sign $-1$
Analytic cond. $84.0307$
Root an. cond. $3.02767$
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 + 7-s + 8-s − 2·9-s + 5·11-s + 14-s + 16-s − 2·18-s + 5·22-s − 8·23-s − 8·25-s + 28-s − 12·29-s + 32-s − 2·36-s + 2·37-s + 43-s + 5·44-s − 8·46-s − 6·49-s − 8·50-s + 3·53-s + 56-s − 12·58-s − 2·63-s + 64-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 2/3·9-s + 1.50·11-s + 0.267·14-s + 1/4·16-s − 0.471·18-s + 1.06·22-s − 1.66·23-s − 8/5·25-s + 0.188·28-s − 2.22·29-s + 0.176·32-s − 1/3·36-s + 0.328·37-s + 0.152·43-s + 0.753·44-s − 1.17·46-s − 6/7·49-s − 1.13·50-s + 0.412·53-s + 0.133·56-s − 1.57·58-s − 0.251·63-s + 1/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1317904\)    =    \(2^{4} \cdot 7^{2} \cdot 41^{2}\)
Sign: $-1$
Analytic conductor: \(84.0307\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1317904} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1317904,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - T + p T^{2} \)
41$C_2$ \( 1 + T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
13$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 29 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \)
59$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 12 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 7 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 103 T^{2} + p^{2} T^{4} \)
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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

−7.70188032956250383472357718174, −7.31659814379934817562700045890, −6.82520071802811922742567132506, −6.30976382521034308355138091266, −5.90486080300567373965680184812, −5.66863666476224430895134878980, −5.20205168631966690076444370145, −4.41748754877673781598733232612, −4.09446932505055114460294912355, −3.69292587461555744645926174264, −3.31683563147727307211984738137, −2.32040457806894073167578856085, −1.99042217290749584909426562590, −1.34269056740568994251551418613, 0, 1.34269056740568994251551418613, 1.99042217290749584909426562590, 2.32040457806894073167578856085, 3.31683563147727307211984738137, 3.69292587461555744645926174264, 4.09446932505055114460294912355, 4.41748754877673781598733232612, 5.20205168631966690076444370145, 5.66863666476224430895134878980, 5.90486080300567373965680184812, 6.30976382521034308355138091266, 6.82520071802811922742567132506, 7.31659814379934817562700045890, 7.70188032956250383472357718174

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