Properties

Label 2-3750-1.1-c1-0-39
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 + 2.61·7-s + 8-s + 9-s − 0.285·11-s + 12-s − 2.80·13-s + 2.61·14-s + 16-s − 6.49·17-s + 18-s − 0.443·19-s + 2.61·21-s − 0.285·22-s + 6.52·23-s + 24-s − 2.80·26-s + 27-s + 2.61·28-s + 7.25·29-s + 3.62·31-s + 32-s − 0.285·33-s − 6.49·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.990·7-s + 0.353·8-s + 0.333·9-s − 0.0861·11-s + 0.288·12-s − 0.778·13-s + 0.700·14-s + 0.250·16-s − 1.57·17-s + 0.235·18-s − 0.101·19-s + 0.571·21-s − 0.0609·22-s + 1.36·23-s + 0.204·24-s − 0.550·26-s + 0.192·27-s + 0.495·28-s + 1.34·29-s + 0.651·31-s + 0.176·32-s − 0.0497·33-s − 1.11·34-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

\(L(1)\) \(\approx\) \(4.227164896\)
\(L(\frac12)\) \(\approx\) \(4.227164896\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.61T + 7T^{2} \)
11 \( 1 + 0.285T + 11T^{2} \)
13 \( 1 + 2.80T + 13T^{2} \)
17 \( 1 + 6.49T + 17T^{2} \)
19 \( 1 + 0.443T + 19T^{2} \)
23 \( 1 - 6.52T + 23T^{2} \)
29 \( 1 - 7.25T + 29T^{2} \)
31 \( 1 - 3.62T + 31T^{2} \)
37 \( 1 - 4.53T + 37T^{2} \)
41 \( 1 - 5.32T + 41T^{2} \)
43 \( 1 - 8.05T + 43T^{2} \)
47 \( 1 - 8.99T + 47T^{2} \)
53 \( 1 - 13.9T + 53T^{2} \)
59 \( 1 + 3.74T + 59T^{2} \)
61 \( 1 - 13.0T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 + 4.81T + 71T^{2} \)
73 \( 1 + 11.7T + 73T^{2} \)
79 \( 1 + 7.81T + 79T^{2} \)
83 \( 1 + 5.16T + 83T^{2} \)
89 \( 1 + 6.16T + 89T^{2} \)
97 \( 1 - 5.45T + 97T^{2} \)
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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

−8.629075728181368871011093692371, −7.58958958804389701947113105255, −7.15155297136461307824475344628, −6.29132616803972599959314871129, −5.31524249985541148302063014554, −4.51030152277949113166352564719, −4.20376139973730182534521663757, −2.72553842351765830521694239079, −2.41702783951563068470337747977, −1.11126194517027649597328244034, 1.11126194517027649597328244034, 2.41702783951563068470337747977, 2.72553842351765830521694239079, 4.20376139973730182534521663757, 4.51030152277949113166352564719, 5.31524249985541148302063014554, 6.29132616803972599959314871129, 7.15155297136461307824475344628, 7.58958958804389701947113105255, 8.629075728181368871011093692371

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