Properties

Label 2-4650-1.1-c1-0-39
Degree $2$
Conductor $4650$
Sign $1$
Analytic cond. $37.1304$
Root an. cond. $6.09347$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s + 3.12·7-s − 8-s + 9-s + 1.56·11-s + 12-s − 3.56·13-s − 3.12·14-s + 16-s + 6.68·17-s − 18-s + 1.56·19-s + 3.12·21-s − 1.56·22-s + 8·23-s − 24-s + 3.56·26-s + 27-s + 3.12·28-s + 1.12·29-s − 31-s − 32-s + 1.56·33-s − 6.68·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s + 1.18·7-s − 0.353·8-s + 0.333·9-s + 0.470·11-s + 0.288·12-s − 0.987·13-s − 0.834·14-s + 0.250·16-s + 1.62·17-s − 0.235·18-s + 0.358·19-s + 0.681·21-s − 0.332·22-s + 1.66·23-s − 0.204·24-s + 0.698·26-s + 0.192·27-s + 0.590·28-s + 0.208·29-s − 0.179·31-s − 0.176·32-s + 0.271·33-s − 1.14·34-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4650,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.289322779\)
\(L(\frac12)\) \(\approx\) \(2.289322779\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.12T + 7T^{2} \)
11 \( 1 - 1.56T + 11T^{2} \)
13 \( 1 + 3.56T + 13T^{2} \)
17 \( 1 - 6.68T + 17T^{2} \)
19 \( 1 - 1.56T + 19T^{2} \)
23 \( 1 - 8T + 23T^{2} \)
29 \( 1 - 1.12T + 29T^{2} \)
37 \( 1 - 8.24T + 37T^{2} \)
41 \( 1 + 12.2T + 41T^{2} \)
43 \( 1 - 0.876T + 43T^{2} \)
47 \( 1 + 8.68T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 7.12T + 59T^{2} \)
61 \( 1 - 3.56T + 61T^{2} \)
67 \( 1 + 12.6T + 67T^{2} \)
71 \( 1 + 5.56T + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 - 8.68T + 79T^{2} \)
83 \( 1 - 7.80T + 83T^{2} \)
89 \( 1 - 16.2T + 89T^{2} \)
97 \( 1 + 5.80T + 97T^{2} \)
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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

−8.360719752039663434630925423786, −7.54495736541336646693849122541, −7.33111759515216688727698393869, −6.30185458533467757363391808862, −5.19978548564065827819494587913, −4.76707196367086176918537659680, −3.52671922712084582984837739607, −2.79867459064052732819003619215, −1.73938275174950761369444365886, −0.989027767179591177449557370591, 0.989027767179591177449557370591, 1.73938275174950761369444365886, 2.79867459064052732819003619215, 3.52671922712084582984837739607, 4.76707196367086176918537659680, 5.19978548564065827819494587913, 6.30185458533467757363391808862, 7.33111759515216688727698393869, 7.54495736541336646693849122541, 8.360719752039663434630925423786

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