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Properties

Label 2-3150-105.89-c1-0-21
Degree $2$
Conductor $3150$
Sign $0.228 - 0.973i$
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 + (2.09 + 1.62i)7-s + 0.999·8-s + (2.59 − 1.5i)11-s − 2.44·13-s + (−2.44 + 0.999i)14-s + (−0.5 + 0.866i)16-s + (0.878 − 0.507i)17-s + (0.878 + 0.507i)19-s + 3i·22-s + (−2.12 + 3.67i)23-s + (1.22 − 2.12i)26-s + (0.358 − 2.62i)28-s + 1.24i·29-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.790 + 0.612i)7-s + 0.353·8-s + (0.783 − 0.452i)11-s − 0.679·13-s + (−0.654 + 0.267i)14-s + (−0.125 + 0.216i)16-s + (0.213 − 0.123i)17-s + (0.201 + 0.116i)19-s + 0.639i·22-s + (−0.442 + 0.766i)23-s + (0.240 − 0.416i)26-s + (0.0677 − 0.495i)28-s + 0.230i·29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.228 - 0.973i$
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: \((2,\ 3150,\ (\ :1/2),\ 0.228 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.644675835\)
\(L(\frac12)\) \(\approx\) \(1.644675835\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.09 - 1.62i)T \)
good11 \( 1 + (-2.59 + 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.44T + 13T^{2} \)
17 \( 1 + (-0.878 + 0.507i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.878 - 0.507i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.12 - 3.67i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.24iT - 29T^{2} \)
31 \( 1 + (-4.86 + 2.80i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.13 - 4.12i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.02T + 41T^{2} \)
43 \( 1 - 8.24iT - 43T^{2} \)
47 \( 1 + (0.878 + 0.507i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.621 + 1.07i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.76 + 9.98i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.12 - 2.95i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.66 + 5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.2iT - 71T^{2} \)
73 \( 1 + (-4.18 - 7.24i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.62 - 9.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.16iT - 83T^{2} \)
89 \( 1 + (-5.19 + 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.76T + 97T^{2} \)
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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

−8.687532341010461077304487866902, −8.067750751867389917472192953787, −7.53080131311968654012347950355, −6.52313153338835365349580616785, −5.93314189287201513956381316622, −5.08670018084250669089189864930, −4.42515889399154306717096800822, −3.26786958719261464420846891917, −2.10038395458285428453921527112, −1.02623631116594382939961784368, 0.71519075067656998607478234768, 1.76695826659496434204368133450, 2.66986614817986451806889984468, 3.88506517240332250535992862771, 4.43622108583776007713647129291, 5.25418191385209623052788802132, 6.41731446377581034458950632369, 7.21453715079056865703804010861, 7.83597353992173425551633453127, 8.578587231776324975930689110589

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