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Properties

Label 2-950-95.32-c1-0-13
Degree $2$
Conductor $950$
Sign $-0.309 - 0.950i$
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  + (−0.0871 + 0.996i)2-s + (0.794 + 1.70i)3-s + (−0.984 − 0.173i)4-s + (−1.76 + 0.642i)6-s + (3.18 − 0.852i)7-s + (0.258 − 0.965i)8-s + (−0.342 + 0.407i)9-s + (0.892 + 1.54i)11-s + (−0.486 − 1.81i)12-s + (0.0794 + 0.0370i)13-s + (0.571 + 3.24i)14-s + (0.939 + 0.342i)16-s + (4.66 + 0.407i)17-s + (−0.376 − 0.376i)18-s + (−2.36 + 3.66i)19-s + ⋯
L(s)  = 1  + (−0.0616 + 0.704i)2-s + (0.458 + 0.983i)3-s + (−0.492 − 0.0868i)4-s + (−0.720 + 0.262i)6-s + (1.20 − 0.322i)7-s + (0.0915 − 0.341i)8-s + (−0.114 + 0.135i)9-s + (0.269 + 0.465i)11-s + (−0.140 − 0.524i)12-s + (0.0220 + 0.0102i)13-s + (0.152 + 0.866i)14-s + (0.234 + 0.0855i)16-s + (1.13 + 0.0989i)17-s + (−0.0886 − 0.0886i)18-s + (−0.542 + 0.840i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.309 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.309 - 0.950i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (507, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.309 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19267 + 1.64313i\)
\(L(\frac12)\) \(\approx\) \(1.19267 + 1.64313i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0871 - 0.996i)T \)
5 \( 1 \)
19 \( 1 + (2.36 - 3.66i)T \)
good3 \( 1 + (-0.794 - 1.70i)T + (-1.92 + 2.29i)T^{2} \)
7 \( 1 + (-3.18 + 0.852i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.892 - 1.54i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.0794 - 0.0370i)T + (8.35 + 9.95i)T^{2} \)
17 \( 1 + (-4.66 - 0.407i)T + (16.7 + 2.95i)T^{2} \)
23 \( 1 + (2.70 + 3.86i)T + (-7.86 + 21.6i)T^{2} \)
29 \( 1 + (-1.51 - 1.27i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-6.29 - 3.63i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.443 + 0.443i)T - 37iT^{2} \)
41 \( 1 + (1.43 - 3.95i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (4.66 + 3.26i)T + (14.7 + 40.4i)T^{2} \)
47 \( 1 + (-0.0133 - 0.152i)T + (-46.2 + 8.16i)T^{2} \)
53 \( 1 + (3.78 - 2.65i)T + (18.1 - 49.8i)T^{2} \)
59 \( 1 + (10.0 - 8.40i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-2.37 + 13.4i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (6.13 - 0.536i)T + (65.9 - 11.6i)T^{2} \)
71 \( 1 + (2.75 - 0.485i)T + (66.7 - 24.2i)T^{2} \)
73 \( 1 + (-6.99 + 3.26i)T + (46.9 - 55.9i)T^{2} \)
79 \( 1 + (12.5 + 4.55i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-1.33 - 4.98i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-6.11 + 2.22i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-0.728 + 8.32i)T + (-95.5 - 16.8i)T^{2} \)
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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

−10.18142494598281215929052007113, −9.463586902016316331495052761855, −8.395419368321502637534229085325, −8.066919148483382271732453276602, −6.97880555697151244880949514246, −5.95154910122924956271497484154, −4.76707554379160461727549089594, −4.36553800580675056472738666806, −3.26915218263765516173372061723, −1.49834088452996495968786419784, 1.11953592690716264957291365351, 2.03102550420263836819897085180, 3.03938490443325731511666768454, 4.36291727222569100843566335156, 5.31817364764549083973918674317, 6.43231639085275216820842213114, 7.62196936211030960813363951566, 8.103450672385417283497549660208, 8.794961708342379835689864189218, 9.819591269137585642733075861722

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