Properties

Label 2-4641-1.1-c1-0-73
Degree $2$
Conductor $4641$
Sign $-1$
Analytic cond. $37.0585$
Root an. cond. $6.08757$
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 − 3-s − 4-s − 2·5-s + 6-s − 7-s + 3·8-s + 9-s + 2·10-s − 4·11-s + 12-s + 13-s + 14-s + 2·15-s − 16-s + 17-s − 18-s − 4·19-s + 2·20-s + 21-s + 4·22-s − 3·24-s − 25-s − 26-s − 27-s + 28-s + 6·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.516·15-s − 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.447·20-s + 0.218·21-s + 0.852·22-s − 0.612·24-s − 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s + 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4641 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4641 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4641\)    =    \(3 \cdot 7 \cdot 13 \cdot 17\)
Sign: $-1$
Analytic conductor: \(37.0585\)
Root analytic conductor: \(6.08757\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4641} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4641,\ (\ :1/2),\ -1)\)

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$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.960900565412759642918177867039, −7.51100578710557827488315749951, −6.60745241365441853014391309087, −5.75307735589360917294707733434, −4.90160227364909605126881294463, −4.28165500022316068332085710726, −3.47496112679539295939607346026, −2.28342198779037191223689949455, −0.864808715135748152743106599523, 0, 0.864808715135748152743106599523, 2.28342198779037191223689949455, 3.47496112679539295939607346026, 4.28165500022316068332085710726, 4.90160227364909605126881294463, 5.75307735589360917294707733434, 6.60745241365441853014391309087, 7.51100578710557827488315749951, 7.960900565412759642918177867039

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