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Properties

Label 2-950-95.3-c1-0-0
Degree $2$
Conductor $950$
Sign $0.216 - 0.976i$
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 + (−3.76 − 1.00i)7-s + (0.258 + 0.965i)8-s + (−0.342 − 0.407i)9-s + (−1.56 + 2.71i)11-s + (−0.486 + 1.81i)12-s + (−1.46 + 0.684i)13-s + (−0.676 + 3.83i)14-s + (0.939 − 0.342i)16-s + (−6.30 + 0.551i)17-s + (−0.376 + 0.376i)18-s + (2.70 + 3.41i)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 + (−1.42 − 0.380i)7-s + (0.0915 + 0.341i)8-s + (−0.114 − 0.135i)9-s + (−0.472 + 0.817i)11-s + (−0.140 + 0.524i)12-s + (−0.407 + 0.189i)13-s + (−0.180 + 1.02i)14-s + (0.234 − 0.0855i)16-s + (−1.52 + 0.133i)17-s + (−0.0886 + 0.0886i)18-s + (0.621 + 0.783i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.216 - 0.976i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.216 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.105915 + 0.0849779i\)
\(L(\frac12)\) \(\approx\) \(0.105915 + 0.0849779i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.70 - 3.41i)T \)
good3 \( 1 + (-0.794 + 1.70i)T + (-1.92 - 2.29i)T^{2} \)
7 \( 1 + (3.76 + 1.00i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.56 - 2.71i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.46 - 0.684i)T + (8.35 - 9.95i)T^{2} \)
17 \( 1 + (6.30 - 0.551i)T + (16.7 - 2.95i)T^{2} \)
23 \( 1 + (0.0168 - 0.0240i)T + (-7.86 - 21.6i)T^{2} \)
29 \( 1 + (-0.496 + 0.416i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (6.55 - 3.78i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.88 + 4.88i)T + 37iT^{2} \)
41 \( 1 + (2.29 + 6.29i)T + (-31.4 + 26.3i)T^{2} \)
43 \( 1 + (-3.26 + 2.28i)T + (14.7 - 40.4i)T^{2} \)
47 \( 1 + (1.16 - 13.3i)T + (-46.2 - 8.16i)T^{2} \)
53 \( 1 + (-0.241 - 0.168i)T + (18.1 + 49.8i)T^{2} \)
59 \( 1 + (5.17 + 4.34i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (0.0829 + 0.470i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (7.83 + 0.685i)T + (65.9 + 11.6i)T^{2} \)
71 \( 1 + (5.21 + 0.918i)T + (66.7 + 24.2i)T^{2} \)
73 \( 1 + (-5.79 - 2.70i)T + (46.9 + 55.9i)T^{2} \)
79 \( 1 + (11.2 - 4.10i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-3.42 + 12.7i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (3.79 + 1.38i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (0.733 + 8.38i)T + (-95.5 + 16.8i)T^{2} \)
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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

−10.23337889641975876142748323063, −9.415094799763359733096076248132, −8.724108640195244631790106773156, −7.47880644759350681550031694156, −7.12392179814583942076922188796, −6.10628469421037208064816230080, −4.75342108597851428125875096551, −3.65466343840630252887074912375, −2.61176820331455800843576488424, −1.71675518680116543360332878288, 0.05738705360603606261527120377, 2.75539984462401996666393337186, 3.50829560195951417230651063384, 4.58395022143734617523468899797, 5.51703316966576074308595083663, 6.48735858181738828423313134933, 7.18502780839100645937241623571, 8.479069802960618282208514788863, 9.067860356386969664970058760129, 9.633492325398058555014582258946

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