Properties

Label 2-3150-105.89-c1-0-16
Degree $2$
Conductor $3150$
Sign $0.973 + 0.228i$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−2.63 − 0.189i)7-s − 0.999·8-s + (−3.44 + 1.99i)11-s − 0.0681·13-s + (−1.48 + 2.19i)14-s + (−0.5 + 0.866i)16-s + (6.34 − 3.66i)17-s + (1.76 + 1.01i)19-s + 3.98i·22-s + (−1.86 + 3.23i)23-s + (−0.0340 + 0.0590i)26-s + (1.15 + 2.38i)28-s − 0.898i·29-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.997 − 0.0716i)7-s − 0.353·8-s + (−1.03 + 0.600i)11-s − 0.0189·13-s + (−0.396 + 0.585i)14-s + (−0.125 + 0.216i)16-s + (1.53 − 0.888i)17-s + (0.404 + 0.233i)19-s + 0.848i·22-s + (−0.389 + 0.673i)23-s + (−0.00668 + 0.0115i)26-s + (0.218 + 0.449i)28-s − 0.166i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.973 + 0.228i$
Analytic conductor: \(25.1528\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3150} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ 0.973 + 0.228i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.548207821\)
\(L(\frac12)\) \(\approx\) \(1.548207821\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (2.63 + 0.189i)T \)
good11 \( 1 + (3.44 - 1.99i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.0681T + 13T^{2} \)
17 \( 1 + (-6.34 + 3.66i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.76 - 1.01i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.86 - 3.23i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.898iT - 29T^{2} \)
31 \( 1 + (4.18 - 2.41i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.52 - 2.03i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.68T + 41T^{2} \)
43 \( 1 - 0.964iT - 43T^{2} \)
47 \( 1 + (-1.43 - 0.830i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.61 - 11.4i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.32 + 9.21i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.51 - 3.76i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.23 + 5.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.93iT - 71T^{2} \)
73 \( 1 + (-5.82 - 10.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.77 + 15.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.3iT - 83T^{2} \)
89 \( 1 + (0.913 - 1.58i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 17.1T + 97T^{2} \)
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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

−8.857177236733062338675585507863, −7.68197147205897028964593719879, −7.34357660792887236478359852870, −6.20106563345206112372022858181, −5.48119997035610861949422629576, −4.85121668838135435060755659605, −3.68258414495251778608112729939, −3.10720839661079125702849744909, −2.21349939972424243821909203790, −0.853752704663457373343975829840, 0.56864169516755822384818064907, 2.35398186055841636489501675538, 3.32630804528499381012007099660, 3.86932681948378655353551182120, 5.15418367884612945975936676618, 5.68591645599612183653716517439, 6.33510021538386171238508603824, 7.19856797753859328436404183672, 7.929074435414858120000880577690, 8.485965031163483589879235062818

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