Properties

Label 2-950-19.17-c1-0-7
Degree $2$
Conductor $950$
Sign $-0.600 - 0.799i$
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  + (−0.173 + 0.984i)2-s + (−0.639 − 0.536i)3-s + (−0.939 − 0.342i)4-s + (0.639 − 0.536i)6-s + (2.09 + 3.62i)7-s + (0.5 − 0.866i)8-s + (−0.399 − 2.26i)9-s + (−2.69 + 4.66i)11-s + (0.417 + 0.723i)12-s + (3.01 − 2.52i)13-s + (−3.93 + 1.43i)14-s + (0.766 + 0.642i)16-s + (0.371 − 2.10i)17-s + 2.30·18-s + (−2.92 + 3.22i)19-s + ⋯
L(s)  = 1  + (−0.122 + 0.696i)2-s + (−0.369 − 0.309i)3-s + (−0.469 − 0.171i)4-s + (0.261 − 0.219i)6-s + (0.790 + 1.36i)7-s + (0.176 − 0.306i)8-s + (−0.133 − 0.755i)9-s + (−0.812 + 1.40i)11-s + (0.120 + 0.208i)12-s + (0.835 − 0.700i)13-s + (−1.05 + 0.382i)14-s + (0.191 + 0.160i)16-s + (0.0901 − 0.511i)17-s + 0.542·18-s + (−0.671 + 0.740i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.600 - 0.799i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.600 - 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.466069 + 0.932931i\)
\(L(\frac12)\) \(\approx\) \(0.466069 + 0.932931i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 - 0.984i)T \)
5 \( 1 \)
19 \( 1 + (2.92 - 3.22i)T \)
good3 \( 1 + (0.639 + 0.536i)T + (0.520 + 2.95i)T^{2} \)
7 \( 1 + (-2.09 - 3.62i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.69 - 4.66i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.01 + 2.52i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.371 + 2.10i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (-2.68 - 0.976i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-1.34 - 7.65i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (2.86 + 4.96i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.84T + 37T^{2} \)
41 \( 1 + (-4.67 - 3.92i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (11.8 - 4.31i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-0.760 - 4.31i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-12.2 - 4.44i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (1.36 - 7.75i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (10.3 + 3.74i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.82 - 10.3i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (12.6 - 4.61i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-2.58 - 2.16i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (3.96 + 3.32i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-4.36 - 7.56i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.52 - 2.12i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-0.150 + 0.851i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.22866663828520709318622453117, −9.278285987725931499551845161704, −8.589946037986013113614994717274, −7.79765708150402808380134562414, −6.96390900498201539347363421769, −5.89772815692284235797628310648, −5.42259201668802981027839411499, −4.44996650575398522266934206099, −2.89792511129944636775804318195, −1.51012635886404558961198129801, 0.55895082021510627576996112749, 1.98586369695712219445047399313, 3.42880869742335138219725758917, 4.34292809026233750700374698151, 5.10431925654265962508265981432, 6.19295616608496446490688138214, 7.40156322515710429291427075737, 8.287031092359184841953212180002, 8.784302572033193760289475478001, 10.23290524957699147668999561875

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