Properties

Label 2-3150-15.14-c2-0-68
Degree $2$
Conductor $3150$
Sign $-0.881 + 0.472i$
Analytic cond. $85.8312$
Root an. cond. $9.26451$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + 2.00·4-s + 2.64i·7-s + 2.82·8-s − 2.32i·11-s − 10.6i·13-s + 3.74i·14-s + 4.00·16-s + 2.70·17-s − 11.0·19-s − 3.29i·22-s − 34.7·23-s − 15.1i·26-s + 5.29i·28-s + 5.82i·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.500·4-s + 0.377i·7-s + 0.353·8-s − 0.211i·11-s − 0.822i·13-s + 0.267i·14-s + 0.250·16-s + 0.159·17-s − 0.584·19-s − 0.149i·22-s − 1.50·23-s − 0.581i·26-s + 0.188i·28-s + 0.200i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.472i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.881 + 0.472i$
Analytic conductor: \(85.8312\)
Root analytic conductor: \(9.26451\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{3150} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1),\ -0.881 + 0.472i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8450955870\)
\(L(\frac12)\) \(\approx\) \(0.8450955870\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 - 2.64iT \)
good11 \( 1 + 2.32iT - 121T^{2} \)
13 \( 1 + 10.6iT - 169T^{2} \)
17 \( 1 - 2.70T + 289T^{2} \)
19 \( 1 + 11.0T + 361T^{2} \)
23 \( 1 + 34.7T + 529T^{2} \)
29 \( 1 - 5.82iT - 841T^{2} \)
31 \( 1 + 15.2T + 961T^{2} \)
37 \( 1 + 0.900iT - 1.36e3T^{2} \)
41 \( 1 + 31.6iT - 1.68e3T^{2} \)
43 \( 1 + 13.0iT - 1.84e3T^{2} \)
47 \( 1 - 17.1T + 2.20e3T^{2} \)
53 \( 1 + 68.1T + 2.80e3T^{2} \)
59 \( 1 - 34.3iT - 3.48e3T^{2} \)
61 \( 1 + 81.5T + 3.72e3T^{2} \)
67 \( 1 + 46.2iT - 4.48e3T^{2} \)
71 \( 1 + 61.3iT - 5.04e3T^{2} \)
73 \( 1 - 20.5iT - 5.32e3T^{2} \)
79 \( 1 + 78.0T + 6.24e3T^{2} \)
83 \( 1 - 28.7T + 6.88e3T^{2} \)
89 \( 1 - 136. iT - 7.92e3T^{2} \)
97 \( 1 - 10.0iT - 9.40e3T^{2} \)
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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

−8.064273442616460951501007870448, −7.47758753640553555517860598424, −6.44691364728802975729422432243, −5.85929464044059792297533217122, −5.19623209345829868544619321115, −4.25258908761991570800709404087, −3.46773092971627077639393651715, −2.57999265592651939084649007244, −1.63270069059436567438691257254, −0.12855695618569474469924825182, 1.47351916552568935610673445882, 2.32659589495495637416740722686, 3.44085016437924028146650111465, 4.23418637093014843690359161545, 4.78275086599152392596412745430, 5.90597437137535189793595111271, 6.39873153019901398961271194606, 7.26899858936126858315864660368, 7.909605483084047736038731290337, 8.764010908384338832771773126267

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