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Properties

Label 2-950-25.6-c1-0-17
Degree $2$
Conductor $950$
Sign $0.190 + 0.981i$
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 + (−1.05 + 0.768i)3-s + (−0.809 + 0.587i)4-s + (−2.23 + 0.144i)5-s + (1.05 + 0.768i)6-s − 0.741·7-s + (0.809 + 0.587i)8-s + (−0.399 + 1.22i)9-s + (0.826 + 2.07i)10-s + (−1.03 − 3.18i)11-s + (0.403 − 1.24i)12-s + (−1.32 + 4.08i)13-s + (0.229 + 0.705i)14-s + (2.24 − 1.86i)15-s + (0.309 − 0.951i)16-s + (0.188 + 0.136i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (−0.610 + 0.443i)3-s + (−0.404 + 0.293i)4-s + (−0.997 + 0.0645i)5-s + (0.431 + 0.313i)6-s − 0.280·7-s + (0.286 + 0.207i)8-s + (−0.133 + 0.409i)9-s + (0.261 + 0.657i)10-s + (−0.312 − 0.961i)11-s + (0.116 − 0.358i)12-s + (−0.368 + 1.13i)13-s + (0.0612 + 0.188i)14-s + (0.580 − 0.482i)15-s + (0.0772 − 0.237i)16-s + (0.0456 + 0.0331i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.190 + 0.981i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.190 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.405116 - 0.333971i\)
\(L(\frac12)\) \(\approx\) \(0.405116 - 0.333971i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 - 0.144i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (1.05 - 0.768i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + 0.741T + 7T^{2} \)
11 \( 1 + (1.03 + 3.18i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.32 - 4.08i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.188 - 0.136i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 + (-0.639 - 1.96i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-4.62 + 3.36i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (3.62 + 2.63i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.0572 + 0.176i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.0574 + 0.176i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 8.70T + 43T^{2} \)
47 \( 1 + (-5.43 + 3.94i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-8.60 + 6.24i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.59 + 4.90i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.49 + 10.7i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (1.57 + 1.14i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-6.06 + 4.40i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.930 - 2.86i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.881 - 0.640i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.50 - 1.82i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (2.14 + 6.59i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (10.0 - 7.30i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.03920528424328136609589225820, −9.121362264340143709491707822692, −8.292471104296763093964100268170, −7.50312587302605952792170425334, −6.39803831902215740035378630594, −5.26742300505596397274914577828, −4.38566130601673192011613168232, −3.56090701674094107654802544852, −2.35286966369154971917084437358, −0.40332561669568633300697586506, 0.881014893253740181941406081240, 2.91634377385752883903438065784, 4.18130532440977705822775368900, 5.15351211838898350974435948806, 6.01712107019720523263323658795, 7.06892546005848938141838030178, 7.42891571231366334448063186182, 8.401650072309053623215016848294, 9.223274461347794399965532606847, 10.29760260636237502958567591828

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