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Properties

Label 2-950-25.6-c1-0-22
Degree $2$
Conductor $950$
Sign $0.249 - 0.968i$
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 + (2.16 − 1.57i)3-s + (−0.809 + 0.587i)4-s + (−0.351 + 2.20i)5-s + (2.16 + 1.57i)6-s + 1.88·7-s + (−0.809 − 0.587i)8-s + (1.28 − 3.96i)9-s + (−2.20 + 0.348i)10-s + (1.99 + 6.12i)11-s + (−0.827 + 2.54i)12-s + (−1.83 + 5.65i)13-s + (0.581 + 1.79i)14-s + (2.71 + 5.33i)15-s + (0.309 − 0.951i)16-s + (−6.32 − 4.59i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (1.25 − 0.908i)3-s + (−0.404 + 0.293i)4-s + (−0.157 + 0.987i)5-s + (0.884 + 0.642i)6-s + 0.711·7-s + (−0.286 − 0.207i)8-s + (0.429 − 1.32i)9-s + (−0.698 + 0.110i)10-s + (0.600 + 1.84i)11-s + (−0.238 + 0.734i)12-s + (−0.509 + 1.56i)13-s + (0.155 + 0.478i)14-s + (0.700 + 1.37i)15-s + (0.0772 − 0.237i)16-s + (−1.53 − 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.249 - 0.968i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.249 - 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04530 + 1.58454i\)
\(L(\frac12)\) \(\approx\) \(2.04530 + 1.58454i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.351 - 2.20i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
good3 \( 1 + (-2.16 + 1.57i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 - 1.88T + 7T^{2} \)
11 \( 1 + (-1.99 - 6.12i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.83 - 5.65i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (6.32 + 4.59i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 + (0.00844 + 0.0259i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-6.89 + 5.01i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.36 - 1.71i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.59 + 4.91i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.84 + 5.68i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 6.45T + 43T^{2} \)
47 \( 1 + (-0.279 + 0.203i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.915 - 0.665i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.80 + 5.56i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.970 + 2.98i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-4.96 - 3.60i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-7.82 + 5.68i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-5.00 - 15.3i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.85 + 2.07i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (11.0 + 8.03i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (3.64 + 11.2i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (11.1 - 8.11i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.761322695159234237574215554550, −9.261903195924450721705381283212, −8.342754534913736296960136289814, −7.38000876096850511582804180962, −7.03666932991525674890953332949, −6.52769874003986235475363033158, −4.67296262706313261546202335023, −4.10752768895032246580913891478, −2.51006638542657174576869724566, −1.99614073185574952168014048560, 1.04492829109742873732496476997, 2.60436246154550298802699312617, 3.47475332830140415980877806269, 4.33869429220341186533966144942, 5.06776804675421973755910465300, 6.14523461932941131962485363539, 8.104606818080487476592012965578, 8.349966826818637077802291404326, 8.933437379038572131502308665977, 9.790417810272055469253626696702

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