Properties

Label 2-3750-1.1-c1-0-47
Degree $2$
Conductor $3750$
Sign $1$
Analytic cond. $29.9439$
Root an. cond. $5.47210$
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 + 4.63·7-s + 8-s + 9-s − 2.53·11-s + 12-s − 0.143·13-s + 4.63·14-s + 16-s + 7.49·17-s + 18-s − 6.74·19-s + 4.63·21-s − 2.53·22-s + 1.67·23-s + 24-s − 0.143·26-s + 27-s + 4.63·28-s − 2.25·29-s + 1.00·31-s + 32-s − 2.53·33-s + 7.49·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s + 1.75·7-s + 0.353·8-s + 0.333·9-s − 0.765·11-s + 0.288·12-s − 0.0399·13-s + 1.23·14-s + 0.250·16-s + 1.81·17-s + 0.235·18-s − 1.54·19-s + 1.01·21-s − 0.541·22-s + 0.349·23-s + 0.204·24-s − 0.0282·26-s + 0.192·27-s + 0.875·28-s − 0.418·29-s + 0.180·31-s + 0.176·32-s − 0.442·33-s + 1.28·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3750\)    =    \(2 \cdot 3 \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(29.9439\)
Root analytic conductor: \(5.47210\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3750,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.697673335\)
\(L(\frac12)\) \(\approx\) \(4.697673335\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good7 \( 1 - 4.63T + 7T^{2} \)
11 \( 1 + 2.53T + 11T^{2} \)
13 \( 1 + 0.143T + 13T^{2} \)
17 \( 1 - 7.49T + 17T^{2} \)
19 \( 1 + 6.74T + 19T^{2} \)
23 \( 1 - 1.67T + 23T^{2} \)
29 \( 1 + 2.25T + 29T^{2} \)
31 \( 1 - 1.00T + 31T^{2} \)
37 \( 1 + 0.0889T + 37T^{2} \)
41 \( 1 - 3.07T + 41T^{2} \)
43 \( 1 - 9.02T + 43T^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 + 4.96T + 53T^{2} \)
59 \( 1 - 5.25T + 59T^{2} \)
61 \( 1 + 13.7T + 61T^{2} \)
67 \( 1 + 7.64T + 67T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 - 1.96T + 73T^{2} \)
79 \( 1 - 0.747T + 79T^{2} \)
83 \( 1 + 3.10T + 83T^{2} \)
89 \( 1 - 0.733T + 89T^{2} \)
97 \( 1 + 9.12T + 97T^{2} \)
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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

−8.343010300575420210229572384374, −7.64647335873143161919614427186, −7.38839271200845889054710973147, −6.02445991603327865373596387275, −5.43145607241255404907210982240, −4.61951136496927733007886927569, −4.06047805878659318807625303824, −2.94120791459850276872889670440, −2.15236954669555256172283128100, −1.23271315866488314536839640903, 1.23271315866488314536839640903, 2.15236954669555256172283128100, 2.94120791459850276872889670440, 4.06047805878659318807625303824, 4.61951136496927733007886927569, 5.43145607241255404907210982240, 6.02445991603327865373596387275, 7.38839271200845889054710973147, 7.64647335873143161919614427186, 8.343010300575420210229572384374

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