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Properties

Label 2-950-95.18-c1-0-8
Degree $2$
Conductor $950$
Sign $-0.909 + 0.416i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 + 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.909 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.909 + 0.416i$
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: \((2,\ 950,\ (\ :1/2),\ -0.909 + 0.416i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.208688 - 0.957539i\)
\(L(\frac12)\) \(\approx\) \(0.208688 - 0.957539i\)
\(L(\frac{3}{2})\) 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

−10.56642238643954638217862292909, −9.589449089132250239855414527415, −8.969554105094386909367512095435, −7.970151445141238750810759426494, −6.81459525078991211935885330351, −5.95052345705738386521777363403, −5.60013063372201694706716204017, −4.26670249384941855059089729183, −3.70911620339499284037357987507, −2.17706269997091858012911464330, 0.41743644502349186815771755511, 1.63649369281237884105284118058, 3.24346014869852400240169144006, 4.04302037571813533201803848998, 5.15774787312333403767408253595, 6.38828613216045133978268694074, 6.65742963583841252727368775246, 7.52267448536130595700122682723, 9.141444249627403814201431110602, 9.517317750066619274482629968897

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