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Properties

Label 2-3150-105.59-c1-0-16
Degree $2$
Conductor $3150$
Sign $-0.420 - 0.907i$
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 + (2.54 − 0.717i)7-s − 0.999·8-s + (5.09 + 2.94i)11-s − 4.05·13-s + (1.89 + 1.84i)14-s + (−0.5 − 0.866i)16-s + (−0.371 − 0.214i)17-s + (−5.30 + 3.06i)19-s + 5.88i·22-s + (0.876 + 1.51i)23-s + (−2.02 − 3.51i)26-s + (−0.651 + 2.56i)28-s − 0.0419i·29-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.962 − 0.271i)7-s − 0.353·8-s + (1.53 + 0.886i)11-s − 1.12·13-s + (0.506 + 0.493i)14-s + (−0.125 − 0.216i)16-s + (−0.0901 − 0.0520i)17-s + (−1.21 + 0.703i)19-s + 1.25i·22-s + (0.182 + 0.316i)23-s + (−0.397 − 0.689i)26-s + (−0.123 + 0.484i)28-s − 0.00778i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.420 - 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.420 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.420 - 0.907i$
Analytic conductor: \(25.1528\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3150} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ -0.420 - 0.907i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.248889223\)
\(L(\frac12)\) \(\approx\) \(2.248889223\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.54 + 0.717i)T \)
good11 \( 1 + (-5.09 - 2.94i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.05T + 13T^{2} \)
17 \( 1 + (0.371 + 0.214i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.30 - 3.06i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.876 - 1.51i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.0419iT - 29T^{2} \)
31 \( 1 + (-7.92 - 4.57i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.928 + 0.536i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.61T + 41T^{2} \)
43 \( 1 - 11.0iT - 43T^{2} \)
47 \( 1 + (-0.834 + 0.481i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.57 + 11.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.77 - 11.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.05 + 0.609i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.9 - 6.32i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.54iT - 71T^{2} \)
73 \( 1 + (4.66 - 8.08i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.35 - 9.28i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.1iT - 83T^{2} \)
89 \( 1 + (-3.15 - 5.46i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.59T + 97T^{2} \)
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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

−8.630787163087287878308514090742, −8.205831159210637872225657658915, −7.19608660586464030809029738201, −6.82891238298874999133081517705, −5.96948764069960510687751998392, −4.85174226949993254515536782176, −4.50906559350463151382765634316, −3.68166093235534285899426991828, −2.33692718573476605227200771130, −1.34117783479542577790129930433, 0.64228693282178756204090750774, 1.85427499700746780059124600433, 2.63672923815692698829802049299, 3.78000457973402070149389197114, 4.51243788071756252427499264923, 5.14375576754819487702406784831, 6.18428446944853749234084602199, 6.74125598171235624340164630081, 7.81922905361213594903728596323, 8.693544448605449037103129367981

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