Properties

Label 2-3750-5.4-c1-0-77
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 no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s i·3-s − 4-s + 6-s − 0.533i·7-s i·8-s − 9-s − 1.43·11-s + i·12-s − 6.57i·13-s + 0.533·14-s + 16-s − 0.958i·17-s i·18-s − 0.212·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.201i·7-s − 0.353i·8-s − 0.333·9-s − 0.432·11-s + 0.288i·12-s − 1.82i·13-s + 0.142·14-s + 0.250·16-s − 0.232i·17-s − 0.235i·18-s − 0.0488·19-s + ⋯

Functional equation

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

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: $\chi_{3750} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3750,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1324486800\)
\(L(\frac12)\) \(\approx\) \(0.1324486800\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + iT \)
5 \( 1 \)
good7 \( 1 + 0.533iT - 7T^{2} \)
11 \( 1 + 1.43T + 11T^{2} \)
13 \( 1 + 6.57iT - 13T^{2} \)
17 \( 1 + 0.958iT - 17T^{2} \)
19 \( 1 + 0.212T + 19T^{2} \)
23 \( 1 + 3.76iT - 23T^{2} \)
29 \( 1 + 6.19T + 29T^{2} \)
31 \( 1 + 2.33T + 31T^{2} \)
37 \( 1 - 4.06iT - 37T^{2} \)
41 \( 1 + 7.94T + 41T^{2} \)
43 \( 1 - 11.3iT - 43T^{2} \)
47 \( 1 - 10.1iT - 47T^{2} \)
53 \( 1 - 3.23iT - 53T^{2} \)
59 \( 1 + 7.52T + 59T^{2} \)
61 \( 1 - 12.5T + 61T^{2} \)
67 \( 1 + 6.91iT - 67T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 - 13.9iT - 73T^{2} \)
79 \( 1 + 15.5T + 79T^{2} \)
83 \( 1 + 16.3iT - 83T^{2} \)
89 \( 1 + 5.62T + 89T^{2} \)
97 \( 1 + 5.56iT - 97T^{2} \)
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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

−7.990029138845986977007852134010, −7.50647679981004974476426784721, −6.72972298233009220249541506670, −5.90309392268583137521924923618, −5.36245875173617713927156583640, −4.51805200700398386244275409367, −3.36930613789030895565680915405, −2.61670221503362409469691949205, −1.18131703825804560583152672820, −0.04027438930330953810016800009, 1.71552123569408679336538730336, 2.37451573293392408961979710484, 3.71458660913321820402697725232, 3.95888377509033898778124210343, 5.10992728337711569513612064845, 5.57440538881152588443287857132, 6.72103075585304244009171785532, 7.40183346638906597358779898758, 8.510310816406402213690555653623, 8.969748265401595008818854447821

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