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Properties

Label 2-950-95.18-c1-0-16
Degree $2$
Conductor $950$
Sign $-0.0344 - 0.999i$
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  + (0.707 + 0.707i)2-s + (−1.16 + 1.16i)3-s + 1.00i·4-s − 1.64·6-s + (2.19 − 2.19i)7-s + (−0.707 + 0.707i)8-s + 0.304i·9-s + 4.06·11-s + (−1.16 − 1.16i)12-s + (0.238 − 0.238i)13-s + 3.10·14-s − 1.00·16-s + (2.63 − 2.63i)17-s + (−0.215 + 0.215i)18-s + (−0.420 + 4.33i)19-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.670 + 0.670i)3-s + 0.500i·4-s − 0.670·6-s + (0.828 − 0.828i)7-s + (−0.250 + 0.250i)8-s + 0.101i·9-s + 1.22·11-s + (−0.335 − 0.335i)12-s + (0.0660 − 0.0660i)13-s + 0.828·14-s − 0.250·16-s + (0.638 − 0.638i)17-s + (−0.0508 + 0.0508i)18-s + (−0.0965 + 0.995i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0344 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0344 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.0344 - 0.999i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.0344 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33651 + 1.38340i\)
\(L(\frac12)\) \(\approx\) \(1.33651 + 1.38340i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 \)
19 \( 1 + (0.420 - 4.33i)T \)
good3 \( 1 + (1.16 - 1.16i)T - 3iT^{2} \)
7 \( 1 + (-2.19 + 2.19i)T - 7iT^{2} \)
11 \( 1 - 4.06T + 11T^{2} \)
13 \( 1 + (-0.238 + 0.238i)T - 13iT^{2} \)
17 \( 1 + (-2.63 + 2.63i)T - 17iT^{2} \)
23 \( 1 + (-2.60 - 2.60i)T + 23iT^{2} \)
29 \( 1 - 6.78T + 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + (3.07 + 3.07i)T + 37iT^{2} \)
41 \( 1 - 8.82iT - 41T^{2} \)
43 \( 1 + (4.02 + 4.02i)T + 43iT^{2} \)
47 \( 1 + (1.81 - 1.81i)T - 47iT^{2} \)
53 \( 1 + (6.19 - 6.19i)T - 53iT^{2} \)
59 \( 1 + 0.737T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + (-8.99 - 8.99i)T + 67iT^{2} \)
71 \( 1 + 11.3iT - 71T^{2} \)
73 \( 1 + (-3.94 - 3.94i)T + 73iT^{2} \)
79 \( 1 - 2.73T + 79T^{2} \)
83 \( 1 + (10.2 + 10.2i)T + 83iT^{2} \)
89 \( 1 + 6.52T + 89T^{2} \)
97 \( 1 + (-8.40 - 8.40i)T + 97iT^{2} \)
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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

−10.33959263191313844487240802321, −9.550271419604452473415398028809, −8.411929768409545422392486341335, −7.62897669752555082566327629535, −6.75890675503396912346860215176, −5.80273668818943395797648141308, −4.90371843002246011231845908086, −4.31665985841296654076577559217, −3.33589373030399729374343807302, −1.38380192027142072617418954342, 1.01034801530187803945432602054, 2.05039335269396865573705762676, 3.42345788666575676799061057436, 4.60974325243506438330894464944, 5.45393186197473935845224233381, 6.36762890032580440556402826577, 6.91048826202206749510458206192, 8.275572102096194115492179807930, 8.989292643656991240110307664761, 9.926820027690910794213692391674

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