Properties

Label 2-4650-1.1-c1-0-75
Degree $2$
Conductor $4650$
Sign $-1$
Analytic cond. $37.1304$
Root an. cond. $6.09347$
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 − 6-s − 7-s + 8-s + 9-s − 11-s − 12-s + 13-s − 14-s + 16-s − 6·17-s + 18-s + 21-s − 22-s − 24-s + 26-s − 27-s − 28-s + 6·29-s + 31-s + 32-s + 33-s − 6·34-s + 36-s + 5·37-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.218·21-s − 0.213·22-s − 0.204·24-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + 1.11·29-s + 0.179·31-s + 0.176·32-s + 0.174·33-s − 1.02·34-s + 1/6·36-s + 0.821·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4650\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 31\)
Sign: $-1$
Analytic conductor: \(37.1304\)
Root analytic conductor: \(6.09347\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4650,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 \)
31 \( 1 - T \)
good7 \( 1 + T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 5 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 15 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 11 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 5 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−7.84204840906848359884327486965, −6.90133229921728365535751443044, −6.42189291458458688505511587086, −5.83366032476747166300676798636, −4.80327014984321196217000035389, −4.46007839354697637445341339047, −3.38436850756224246282310830134, −2.57482715218498483234227841295, −1.47462250124641680907921594200, 0, 1.47462250124641680907921594200, 2.57482715218498483234227841295, 3.38436850756224246282310830134, 4.46007839354697637445341339047, 4.80327014984321196217000035389, 5.83366032476747166300676798636, 6.42189291458458688505511587086, 6.90133229921728365535751443044, 7.84204840906848359884327486965

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