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Properties

Label 2-950-25.21-c1-0-23
Degree $2$
Conductor $950$
Sign $0.977 + 0.211i$
Analytic cond. $7.58578$
Root an. cond. $2.75423$
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  + (−0.309 + 0.951i)2-s + (−0.591 − 0.430i)3-s + (−0.809 − 0.587i)4-s + (1.69 − 1.45i)5-s + (0.591 − 0.430i)6-s + 1.38·7-s + (0.809 − 0.587i)8-s + (−0.761 − 2.34i)9-s + (0.862 + 2.06i)10-s + (−0.114 + 0.352i)11-s + (0.226 + 0.695i)12-s + (2.21 + 6.83i)13-s + (−0.427 + 1.31i)14-s + (−1.63 + 0.133i)15-s + (0.309 + 0.951i)16-s + (−0.445 + 0.324i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.341 − 0.248i)3-s + (−0.404 − 0.293i)4-s + (0.758 − 0.651i)5-s + (0.241 − 0.175i)6-s + 0.523·7-s + (0.286 − 0.207i)8-s + (−0.253 − 0.781i)9-s + (0.272 + 0.652i)10-s + (−0.0345 + 0.106i)11-s + (0.0652 + 0.200i)12-s + (0.615 + 1.89i)13-s + (−0.114 + 0.351i)14-s + (−0.420 + 0.0345i)15-s + (0.0772 + 0.237i)16-s + (−0.108 + 0.0785i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.977 + 0.211i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.977 + 0.211i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44818 - 0.155081i\)
\(L(\frac12)\) \(\approx\) \(1.44818 - 0.155081i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (-1.69 + 1.45i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (0.591 + 0.430i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 - 1.38T + 7T^{2} \)
11 \( 1 + (0.114 - 0.352i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-2.21 - 6.83i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.445 - 0.324i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 + (-1.88 + 5.80i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-0.126 - 0.0919i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-7.87 + 5.72i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.72 + 5.30i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.58 + 4.86i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 9.38T + 43T^{2} \)
47 \( 1 + (7.53 + 5.47i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-9.74 - 7.08i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.34 - 4.13i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (4.11 - 12.6i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-6.18 + 4.49i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (0.837 + 0.608i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.25 + 6.94i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-3.79 - 2.75i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-13.4 + 9.76i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-4.28 + 13.1i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (9.59 + 6.96i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.746219555326610627020749283033, −8.916734360088753086767077865566, −8.652266507789403368115325546193, −7.32786052777048367226817564926, −6.43731891472603447694096070010, −5.97534416471934647494057617103, −4.83854176140197031048268020803, −4.09406877765744786314332374451, −2.15035788181693845514892181666, −0.918033610906225322018630836118, 1.28111658903089941058829036720, 2.63790545068119470426817478894, 3.42401135619596054251191157618, 5.00442808870805268505242184244, 5.47286795795352651046741621729, 6.56056036906892858605479186401, 7.900923186368228486604432616374, 8.276722796899547834249032453183, 9.608610102075809247058676393960, 10.17673859714717464270523277897

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