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Properties

Label 2-950-19.6-c1-0-30
Degree $2$
Conductor $950$
Sign $-0.861 + 0.508i$
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  + (0.766 − 0.642i)2-s + (0.0808 + 0.0294i)3-s + (0.173 − 0.984i)4-s + (0.0808 − 0.0294i)6-s + (1.91 − 3.32i)7-s + (−0.500 − 0.866i)8-s + (−2.29 − 1.92i)9-s + (−1.81 − 3.15i)11-s + (0.0430 − 0.0744i)12-s + (−5.10 + 1.85i)13-s + (−0.666 − 3.77i)14-s + (−0.939 − 0.342i)16-s + (−3.33 + 2.79i)17-s − 2.99·18-s + (2.88 + 3.26i)19-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (0.0466 + 0.0169i)3-s + (0.0868 − 0.492i)4-s + (0.0329 − 0.0120i)6-s + (0.725 − 1.25i)7-s + (−0.176 − 0.306i)8-s + (−0.764 − 0.641i)9-s + (−0.548 − 0.950i)11-s + (0.0124 − 0.0215i)12-s + (−1.41 + 0.515i)13-s + (−0.178 − 1.01i)14-s + (−0.234 − 0.0855i)16-s + (−0.808 + 0.678i)17-s − 0.705·18-s + (0.661 + 0.750i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 + 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.861 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.861 + 0.508i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.861 + 0.508i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.410510 - 1.50392i\)
\(L(\frac12)\) \(\approx\) \(0.410510 - 1.50392i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
5 \( 1 \)
19 \( 1 + (-2.88 - 3.26i)T \)
good3 \( 1 + (-0.0808 - 0.0294i)T + (2.29 + 1.92i)T^{2} \)
7 \( 1 + (-1.91 + 3.32i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.81 + 3.15i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.10 - 1.85i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (3.33 - 2.79i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-0.631 + 3.58i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-0.814 - 0.683i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-0.846 + 1.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.69T + 37T^{2} \)
41 \( 1 + (-10.5 - 3.84i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.96 + 11.1i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-4.24 - 3.56i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-0.464 + 2.63i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-8.09 + 6.79i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-1.77 + 10.0i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (4.86 + 4.08i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (0.369 + 2.09i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (6.51 + 2.37i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-7.31 - 2.66i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (1.60 - 2.77i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.01 - 1.09i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-13.7 + 11.5i)T + (16.8 - 95.5i)T^{2} \)
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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

−9.929488434415135073727480232480, −8.894525963041280334728344823709, −8.005225699093635300716184369202, −7.11049013034131999485741907461, −6.13891742481115773757109071537, −5.13932517867974629124822173563, −4.26487887510797384119940203917, −3.35792128319872664724497175667, −2.15845107911688296041486268832, −0.56631810796416234061333537063, 2.35334770093055227814938588127, 2.73381195262887712268532114091, 4.65557107740844598622427577231, 5.12493771757861060907356032638, 5.77270477448562226135970468815, 7.23353694828866195405594897845, 7.63338242307300308414816435809, 8.664750769482900112924489199873, 9.338548207908886112832503439469, 10.41893422906761459329249321320

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