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Properties

Label 2-950-95.9-c1-0-21
Degree $2$
Conductor $950$
Sign $-0.385 + 0.922i$
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  + (0.984 − 0.173i)2-s + (−1.90 − 2.27i)3-s + (0.939 − 0.342i)4-s + (−2.27 − 1.90i)6-s + (2.51 + 1.45i)7-s + (0.866 − 0.5i)8-s + (−1.00 + 5.70i)9-s + (−0.688 − 1.19i)11-s + (−2.56 − 1.48i)12-s + (3.44 − 4.11i)13-s + (2.73 + 0.994i)14-s + (0.766 − 0.642i)16-s + (5.22 − 0.920i)17-s + 5.78i·18-s + (−2.17 − 3.77i)19-s + ⋯
L(s)  = 1  + (0.696 − 0.122i)2-s + (−1.10 − 1.31i)3-s + (0.469 − 0.171i)4-s + (−0.927 − 0.777i)6-s + (0.952 + 0.549i)7-s + (0.306 − 0.176i)8-s + (−0.335 + 1.90i)9-s + (−0.207 − 0.359i)11-s + (−0.741 − 0.427i)12-s + (0.956 − 1.14i)13-s + (0.730 + 0.265i)14-s + (0.191 − 0.160i)16-s + (1.26 − 0.223i)17-s + 1.36i·18-s + (−0.497 − 0.867i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.385 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.385 + 0.922i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.385 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01909 - 1.52986i\)
\(L(\frac12)\) \(\approx\) \(1.01909 - 1.52986i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.984 + 0.173i)T \)
5 \( 1 \)
19 \( 1 + (2.17 + 3.77i)T \)
good3 \( 1 + (1.90 + 2.27i)T + (-0.520 + 2.95i)T^{2} \)
7 \( 1 + (-2.51 - 1.45i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.688 + 1.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.44 + 4.11i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-5.22 + 0.920i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (0.546 + 1.50i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.411 + 2.33i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (3.50 - 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.31iT - 37T^{2} \)
41 \( 1 + (7.13 - 5.98i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (1.99 - 5.48i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (-2.41 - 0.425i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (3.63 + 9.98i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (-1.39 - 7.93i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-14.5 + 5.28i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (10.0 + 1.77i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (2.36 + 0.861i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-4.37 - 5.21i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (-6.31 + 5.29i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (0.419 + 0.242i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (9.24 + 7.76i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-9.01 + 1.59i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.21592501901942432248661507963, −8.548408293829980092388283338218, −7.942063575941895680421903854915, −7.06801785090553992554618670820, −6.11475348914157093840714521807, −5.51020990357214265544089189668, −4.90422527532719378982150383229, −3.24845076056255842361859639229, −1.97396473551964153871077682583, −0.862171704049640687991183815086, 1.57023623534908354513730964944, 3.65645826918948496260269854316, 4.13550060349737022917835685104, 5.02501507902305373189785446559, 5.70636968034850225994080074863, 6.56079224129352627495233538566, 7.66538742760171290751340145354, 8.685203460463093343474195421232, 9.841641449218630376980354707934, 10.45617916126995890040860682746

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