Properties

Label 2-3150-105.104-c1-0-35
Degree $2$
Conductor $3150$
Sign $-0.970 + 0.241i$
Analytic cond. $25.1528$
Root an. cond. $5.01526$
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  − 2-s + 4-s + (0.648 − 2.56i)7-s − 8-s + 1.29i·11-s − 3.13·13-s + (−0.648 + 2.56i)14-s + 16-s + 5.53i·17-s + 7.37i·19-s − 1.29i·22-s − 1.83·23-s + 3.13·26-s + (0.648 − 2.56i)28-s + 1.83i·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (0.245 − 0.969i)7-s − 0.353·8-s + 0.390i·11-s − 0.868·13-s + (−0.173 + 0.685i)14-s + 0.250·16-s + 1.34i·17-s + 1.69i·19-s − 0.276i·22-s − 0.382·23-s + 0.613·26-s + (0.122 − 0.484i)28-s + 0.340i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.970 + 0.241i$
Analytic conductor: \(25.1528\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3150} (3149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ -0.970 + 0.241i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2022494175\)
\(L(\frac12)\) \(\approx\) \(0.2022494175\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7 \( 1 + (-0.648 + 2.56i)T \)
good11 \( 1 - 1.29iT - 11T^{2} \)
13 \( 1 + 3.13T + 13T^{2} \)
17 \( 1 - 5.53iT - 17T^{2} \)
19 \( 1 - 7.37iT - 19T^{2} \)
23 \( 1 + 1.83T + 23T^{2} \)
29 \( 1 - 1.83iT - 29T^{2} \)
31 \( 1 + 10.4iT - 31T^{2} \)
37 \( 1 + 10.6iT - 37T^{2} \)
41 \( 1 - 3.13T + 41T^{2} \)
43 \( 1 + 3.53iT - 43T^{2} \)
47 \( 1 + 10.7iT - 47T^{2} \)
53 \( 1 + 4.42T + 53T^{2} \)
59 \( 1 + 7.18T + 59T^{2} \)
61 \( 1 - 4.88iT - 61T^{2} \)
67 \( 1 + 9.79iT - 67T^{2} \)
71 \( 1 - 7.37iT - 71T^{2} \)
73 \( 1 + 3.40T + 73T^{2} \)
79 \( 1 + 9.01T + 79T^{2} \)
83 \( 1 - 6.26iT - 83T^{2} \)
89 \( 1 - 7.94T + 89T^{2} \)
97 \( 1 + 8.09T + 97T^{2} \)
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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

−8.096511392413405717724126341366, −7.76870417712743952015042485688, −7.07449549057329479235050948232, −6.15769458811971019022711453216, −5.44485915029638928470307586576, −4.17151226634443720677987034365, −3.74756954378743711643184366801, −2.27448640888268070174712476174, −1.50932956047106054712998559153, −0.079188706485313008974897166107, 1.33062058394652764360774399003, 2.70130752775983548464646715240, 2.92090148488734252992407009456, 4.70309168254699696766387623113, 5.08918244875897183488540201689, 6.18561848249280045863856634852, 6.86576591277872111764710143696, 7.63588246164962866650905289321, 8.358777577143606505165251620682, 9.170746385507936200932940365273

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