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Properties

Label 2-950-25.21-c1-0-31
Degree $2$
Conductor $950$
Sign $-0.484 + 0.874i$
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  + (0.309 − 0.951i)2-s + (0.521 + 0.378i)3-s + (−0.809 − 0.587i)4-s + (−0.491 + 2.18i)5-s + (0.521 − 0.378i)6-s − 2.52·7-s + (−0.809 + 0.587i)8-s + (−0.798 − 2.45i)9-s + (1.92 + 1.14i)10-s + (0.688 − 2.11i)11-s + (−0.199 − 0.613i)12-s + (−0.343 − 1.05i)13-s + (−0.781 + 2.40i)14-s + (−1.08 + 0.951i)15-s + (0.309 + 0.951i)16-s + (5.55 − 4.03i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (0.301 + 0.218i)3-s + (−0.404 − 0.293i)4-s + (−0.219 + 0.975i)5-s + (0.212 − 0.154i)6-s − 0.956·7-s + (−0.286 + 0.207i)8-s + (−0.266 − 0.819i)9-s + (0.608 + 0.360i)10-s + (0.207 − 0.638i)11-s + (−0.0575 − 0.176i)12-s + (−0.0952 − 0.293i)13-s + (−0.208 + 0.643i)14-s + (−0.279 + 0.245i)15-s + (0.0772 + 0.237i)16-s + (1.34 − 0.978i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 + 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.484 + 0.874i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.484 + 0.874i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.612612 - 1.03945i\)
\(L(\frac12)\) \(\approx\) \(0.612612 - 1.03945i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (0.491 - 2.18i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (-0.521 - 0.378i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + 2.52T + 7T^{2} \)
11 \( 1 + (-0.688 + 2.11i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.343 + 1.05i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-5.55 + 4.03i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 + (-0.471 + 1.45i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (5.23 + 3.80i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.0336 - 0.0244i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.11 + 3.41i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.40 + 4.31i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.28T + 43T^{2} \)
47 \( 1 + (1.11 + 0.813i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.355 - 0.258i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.99 + 12.2i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.54 - 7.82i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-3.05 + 2.22i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-13.4 - 9.76i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.52 + 13.9i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (1.48 + 1.07i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (8.43 - 6.12i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (3.76 - 11.5i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (12.4 + 9.01i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.627011704925234206177079431416, −9.426243034107965847884954989197, −8.169675992666122267743203574223, −7.16517818897223745272852269898, −6.24831290095765988227647778015, −5.47073264061361636835976229416, −3.86183069760243364161459475345, −3.32785195550131598575839866131, −2.60742841168933381191311159708, −0.51580345990424214524871048322, 1.56595437652287481005046068345, 3.19094706058657643659399205420, 4.17220022856253712162730102853, 5.21678696148416284020466125732, 5.91740000053705661320292571279, 7.06277224330677324360566522382, 7.79720770617070460942282786640, 8.474589799734658614238521134056, 9.393432011111530470296860559048, 9.965288447833867109967669208324

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