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Properties

Label 2-950-19.4-c1-0-10
Degree $2$
Conductor $950$
Sign $-0.0462 - 0.998i$
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  + (0.939 + 0.342i)2-s + (0.0710 + 0.402i)3-s + (0.766 + 0.642i)4-s + (−0.0710 + 0.402i)6-s + (−1.15 + 1.99i)7-s + (0.500 + 0.866i)8-s + (2.66 − 0.968i)9-s + (1.32 + 2.29i)11-s + (−0.204 + 0.354i)12-s + (−0.885 + 5.02i)13-s + (−1.76 + 1.47i)14-s + (0.173 + 0.984i)16-s + (−2.62 − 0.955i)17-s + 2.83·18-s + (−1.11 − 4.21i)19-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (0.0409 + 0.232i)3-s + (0.383 + 0.321i)4-s + (−0.0289 + 0.164i)6-s + (−0.434 + 0.753i)7-s + (0.176 + 0.306i)8-s + (0.887 − 0.322i)9-s + (0.399 + 0.691i)11-s + (−0.0590 + 0.102i)12-s + (−0.245 + 1.39i)13-s + (−0.471 + 0.395i)14-s + (0.0434 + 0.246i)16-s + (−0.636 − 0.231i)17-s + 0.667·18-s + (−0.256 − 0.966i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0462 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0462 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.0462 - 0.998i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.0462 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58724 + 1.66238i\)
\(L(\frac12)\) \(\approx\) \(1.58724 + 1.66238i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 - 0.342i)T \)
5 \( 1 \)
19 \( 1 + (1.11 + 4.21i)T \)
good3 \( 1 + (-0.0710 - 0.402i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (1.15 - 1.99i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.32 - 2.29i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.885 - 5.02i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (2.62 + 0.955i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (0.896 + 0.751i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-4.18 + 1.52i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (4.11 - 7.12i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.99T + 37T^{2} \)
41 \( 1 + (-1.97 - 11.2i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-1.18 + 0.990i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-9.92 + 3.61i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-3.24 - 2.72i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (10.5 + 3.83i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-2.53 - 2.12i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (7.91 - 2.88i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (0.239 - 0.200i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (2.66 + 15.1i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (1.52 + 8.67i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (0.887 - 1.53i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.12 + 6.39i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-2.46 - 0.896i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.19512069882706302769110572271, −9.230022228476280666894261530620, −8.897067300211614159069674911760, −7.36412553078199782619189936647, −6.78154309951343413206488228416, −6.07887902922476530108944876658, −4.62294422887965558863343802238, −4.38602522535057332452433329097, −2.97878518109397887856343331544, −1.85412438800418666048524018521, 0.893137890075779955246135330330, 2.34652104300422554066271190700, 3.63034742256945371020684107047, 4.24620502067461427589570138709, 5.52277345384952584201778448028, 6.27491195977933194457077846476, 7.28154397387877069563892367127, 7.892033186529811059427165532220, 9.070063654900111100476585116604, 10.30979791737100828817488782717

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