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Properties

Label 2-950-19.16-c1-0-0
Degree $2$
Conductor $950$
Sign $-0.813 - 0.581i$
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 + (−0.0801 + 0.0291i)3-s + (0.173 + 0.984i)4-s + (0.0801 + 0.0291i)6-s + (0.920 + 1.59i)7-s + (0.500 − 0.866i)8-s + (−2.29 + 1.92i)9-s + (−1.21 + 2.09i)11-s + (−0.0426 − 0.0738i)12-s + (2.84 + 1.03i)13-s + (0.319 − 1.81i)14-s + (−0.939 + 0.342i)16-s + (−5.22 − 4.38i)17-s + 2.99·18-s + (−3.15 + 3.00i)19-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (−0.0462 + 0.0168i)3-s + (0.0868 + 0.492i)4-s + (0.0327 + 0.0119i)6-s + (0.347 + 0.602i)7-s + (0.176 − 0.306i)8-s + (−0.764 + 0.641i)9-s + (−0.364 + 0.632i)11-s + (−0.0123 − 0.0213i)12-s + (0.790 + 0.287i)13-s + (0.0854 − 0.484i)14-s + (−0.234 + 0.0855i)16-s + (−1.26 − 1.06i)17-s + 0.705·18-s + (−0.724 + 0.689i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.813 - 0.581i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.813 - 0.581i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0938523 + 0.292547i\)
\(L(\frac12)\) \(\approx\) \(0.0938523 + 0.292547i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.15 - 3.00i)T \)
good3 \( 1 + (0.0801 - 0.0291i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (-0.920 - 1.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.21 - 2.09i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.84 - 1.03i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (5.22 + 4.38i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (1.39 + 7.90i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (7.77 - 6.52i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (2.41 + 4.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.202T + 37T^{2} \)
41 \( 1 + (-5.41 + 1.97i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.365 - 2.07i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (7.53 - 6.32i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (1.08 + 6.17i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (4.62 + 3.87i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.741 - 4.20i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (2.39 - 2.00i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.94 - 11.0i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (7.33 - 2.67i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (9.87 - 3.59i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-6.37 - 11.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.21 + 0.442i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-6.28 - 5.27i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.54119535115238779618768683683, −9.427270756596744902378810587024, −8.724592308639456756946427164976, −8.156038607356976097104250822645, −7.13541480542963939179165874968, −6.12162298365629018882196283000, −5.06986691270349323747708128819, −4.11911771010718638250869449674, −2.64097221720868259014396854651, −1.94326437185619496097337709831, 0.16189684895448566955701916579, 1.74624379562518576642864751090, 3.33446152058586896933779205236, 4.35270266417163645732251417038, 5.74035231627136374138319757231, 6.17214069591325466494414481563, 7.26648598640485996790165084417, 8.124645746588032059130190836936, 8.794257329306958391485531669328, 9.473404882082165815125261629785

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