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Properties

Label 2-950-475.111-c1-0-44
Degree $2$
Conductor $950$
Sign $-0.973 - 0.230i$
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  + (−0.997 − 0.0697i)2-s + (−0.0644 − 1.84i)3-s + (0.990 + 0.139i)4-s + (0.0499 − 2.23i)5-s + (−0.0644 + 1.84i)6-s + (−1.02 − 1.77i)7-s + (−0.978 − 0.207i)8-s + (−0.410 + 0.0287i)9-s + (−0.205 + 2.22i)10-s + (−0.327 − 3.11i)11-s + (0.193 − 1.83i)12-s + (−1.21 − 1.80i)13-s + (0.900 + 1.84i)14-s + (−4.13 + 0.0518i)15-s + (0.961 + 0.275i)16-s + (−1.19 + 2.95i)17-s + ⋯
L(s)  = 1  + (−0.705 − 0.0493i)2-s + (−0.0372 − 1.06i)3-s + (0.495 + 0.0695i)4-s + (0.0223 − 0.999i)5-s + (−0.0263 + 0.753i)6-s + (−0.388 − 0.672i)7-s + (−0.345 − 0.0735i)8-s + (−0.136 + 0.00957i)9-s + (−0.0650 + 0.704i)10-s + (−0.0986 − 0.938i)11-s + (0.0557 − 0.530i)12-s + (−0.338 − 0.501i)13-s + (0.240 + 0.493i)14-s + (−1.06 + 0.0133i)15-s + (0.240 + 0.0689i)16-s + (−0.289 + 0.717i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.973 - 0.230i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.973 - 0.230i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0998078 + 0.853297i\)
\(L(\frac12)\) \(\approx\) \(0.0998078 + 0.853297i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.997 + 0.0697i)T \)
5 \( 1 + (-0.0499 + 2.23i)T \)
19 \( 1 + (-3.28 + 2.86i)T \)
good3 \( 1 + (0.0644 + 1.84i)T + (-2.99 + 0.209i)T^{2} \)
7 \( 1 + (1.02 + 1.77i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.327 + 3.11i)T + (-10.7 + 2.28i)T^{2} \)
13 \( 1 + (1.21 + 1.80i)T + (-4.86 + 12.0i)T^{2} \)
17 \( 1 + (1.19 - 2.95i)T + (-12.2 - 11.8i)T^{2} \)
23 \( 1 + (1.40 + 1.35i)T + (0.802 + 22.9i)T^{2} \)
29 \( 1 + (-2.22 - 5.51i)T + (-20.8 + 20.1i)T^{2} \)
31 \( 1 + (-2.20 + 2.44i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (-2.34 - 1.70i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.74 - 0.498i)T + (34.7 + 21.7i)T^{2} \)
43 \( 1 + (0.386 - 2.18i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (0.412 + 1.02i)T + (-33.8 + 32.6i)T^{2} \)
53 \( 1 + (5.81 + 0.817i)T + (50.9 + 14.6i)T^{2} \)
59 \( 1 + (-3.23 - 12.9i)T + (-52.0 + 27.6i)T^{2} \)
61 \( 1 + (-0.111 - 0.107i)T + (2.12 + 60.9i)T^{2} \)
67 \( 1 + (-4.85 - 3.03i)T + (29.3 + 60.2i)T^{2} \)
71 \( 1 + (-0.811 - 0.431i)T + (39.7 + 58.8i)T^{2} \)
73 \( 1 + (-0.434 + 0.644i)T + (-27.3 - 67.6i)T^{2} \)
79 \( 1 + (-0.00261 - 0.0749i)T + (-78.8 + 5.51i)T^{2} \)
83 \( 1 + (6.18 - 6.86i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 + (-1.87 + 0.536i)T + (75.4 - 47.1i)T^{2} \)
97 \( 1 + (-0.158 + 0.0990i)T + (42.5 - 87.1i)T^{2} \)
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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

−9.542777968079478093863226565474, −8.551826565291515262819585533730, −8.014928027744573655052688856223, −7.18491760929289058612421294811, −6.39464327813658196088814197832, −5.45339024308540531928007892051, −4.14715767870134440391212662681, −2.79750753288745115739559889929, −1.37127766020497541197119230630, −0.53486011152825515810215713669, 2.09621978154659616153119395075, 3.09659606284956465631880453083, 4.21660757300371618811611508481, 5.29402438417579991436841485266, 6.36203276342964489007482311291, 7.16631102964051588247253747840, 7.969254544139273606216663114707, 9.291671288175695558862206865738, 9.672441394217085084114080255807, 10.17290651536101108803384647180

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