Properties

Label 2-3750-1.1-c1-0-51
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 + 2.70·7-s + 8-s + 9-s + 5.62·11-s + 12-s − 4.84·13-s + 2.70·14-s + 16-s + 1.04·17-s + 18-s + 8.38·19-s + 2.70·21-s + 5.62·22-s − 5.29·23-s + 24-s − 4.84·26-s + 27-s + 2.70·28-s + 4.19·29-s − 5.58·31-s + 32-s + 5.62·33-s + 1.04·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s + 1.02·7-s + 0.353·8-s + 0.333·9-s + 1.69·11-s + 0.288·12-s − 1.34·13-s + 0.724·14-s + 0.250·16-s + 0.253·17-s + 0.235·18-s + 1.92·19-s + 0.591·21-s + 1.19·22-s − 1.10·23-s + 0.204·24-s − 0.949·26-s + 0.192·27-s + 0.511·28-s + 0.778·29-s − 1.00·31-s + 0.176·32-s + 0.978·33-s + 0.179·34-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

\(L(1)\) \(\approx\) \(4.718970751\)
\(L(\frac12)\) \(\approx\) \(4.718970751\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.70T + 7T^{2} \)
11 \( 1 - 5.62T + 11T^{2} \)
13 \( 1 + 4.84T + 13T^{2} \)
17 \( 1 - 1.04T + 17T^{2} \)
19 \( 1 - 8.38T + 19T^{2} \)
23 \( 1 + 5.29T + 23T^{2} \)
29 \( 1 - 4.19T + 29T^{2} \)
31 \( 1 + 5.58T + 31T^{2} \)
37 \( 1 + 2.99T + 37T^{2} \)
41 \( 1 + 1.57T + 41T^{2} \)
43 \( 1 - 3.29T + 43T^{2} \)
47 \( 1 + 3.07T + 47T^{2} \)
53 \( 1 - 6.17T + 53T^{2} \)
59 \( 1 + 14.1T + 59T^{2} \)
61 \( 1 - 0.270T + 61T^{2} \)
67 \( 1 - 6.30T + 67T^{2} \)
71 \( 1 + 4.53T + 71T^{2} \)
73 \( 1 + 8.16T + 73T^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 + 0.134T + 83T^{2} \)
89 \( 1 - 11.7T + 89T^{2} \)
97 \( 1 + 2.64T + 97T^{2} \)
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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

−8.401679494682803377109974882396, −7.58416843566055506803381546388, −7.20398224564719640445478745411, −6.25877038531537594175937847902, −5.31341024738849373456573279944, −4.69045252013010129432175297452, −3.89143254848788070099668054093, −3.13363114441946245711659483239, −2.03937462931733106993822400725, −1.26083684138652035941074686546, 1.26083684138652035941074686546, 2.03937462931733106993822400725, 3.13363114441946245711659483239, 3.89143254848788070099668054093, 4.69045252013010129432175297452, 5.31341024738849373456573279944, 6.25877038531537594175937847902, 7.20398224564719640445478745411, 7.58416843566055506803381546388, 8.401679494682803377109974882396

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