Properties

Label 2-3150-15.14-c2-0-41
Degree $2$
Conductor $3150$
Sign $0.151 + 0.988i$
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 + 18.6i·13-s − 3.74i·14-s + 4.00·16-s + 12.0·17-s − 30.0·19-s − 3.29i·22-s − 21.5·23-s − 26.4i·26-s + 5.29i·28-s + 23.6i·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 + 1.43i·13-s − 0.267i·14-s + 0.250·16-s + 0.706·17-s − 1.58·19-s − 0.149i·22-s − 0.937·23-s − 1.01i·26-s + 0.188i·28-s + 0.814i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.151 + 0.988i$
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: \((2,\ 3150,\ (\ :1),\ 0.151 + 0.988i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5598764746\)
\(L(\frac12)\) \(\approx\) \(0.5598764746\)
\(L(2)\) 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 \( 1 - 18.6iT - 169T^{2} \)
17 \( 1 - 12.0T + 289T^{2} \)
19 \( 1 + 30.0T + 361T^{2} \)
23 \( 1 + 21.5T + 529T^{2} \)
29 \( 1 - 23.6iT - 841T^{2} \)
31 \( 1 + 55.0T + 961T^{2} \)
37 \( 1 + 38.8iT - 1.36e3T^{2} \)
41 \( 1 + 39.3iT - 1.68e3T^{2} \)
43 \( 1 + 71.8iT - 1.84e3T^{2} \)
47 \( 1 - 80.6T + 2.20e3T^{2} \)
53 \( 1 + 71.1T + 2.80e3T^{2} \)
59 \( 1 - 51.3iT - 3.48e3T^{2} \)
61 \( 1 + 1.95T + 3.72e3T^{2} \)
67 \( 1 - 67.6iT - 4.48e3T^{2} \)
71 \( 1 - 56.1iT - 5.04e3T^{2} \)
73 \( 1 + 126. iT - 5.32e3T^{2} \)
79 \( 1 - 35.7T + 6.24e3T^{2} \)
83 \( 1 - 39.6T + 6.88e3T^{2} \)
89 \( 1 + 56.3iT - 7.92e3T^{2} \)
97 \( 1 + 129. iT - 9.40e3T^{2} \)
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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

−8.666218074834490488463220394842, −7.48623975060821184466115070887, −7.07180401349994645107814259949, −6.13301289569100948352884322216, −5.49727811537413690166471279743, −4.31246383905650296489168556020, −3.61756402226634345587532190726, −2.20730037382501535196588801065, −1.78546442889380614906303049032, −0.19476172534054218785287757565, 0.812333214028260789780106008860, 1.95295200478049039619329132319, 2.99496643088488716374437214181, 3.83222182631220300446224268508, 4.87337927256716485309266192980, 5.91320800969175535790252127798, 6.35356700648301836366332256391, 7.47201549656782014275942730782, 7.997724475841473785413183705812, 8.479598573567606575746729550501

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