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Properties

Label 2-950-19.16-c1-0-21
Degree $2$
Conductor $950$
Sign $0.500 + 0.865i$
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  + (−0.766 − 0.642i)2-s + (1.76 − 0.642i)3-s + (0.173 + 0.984i)4-s + (−1.76 − 0.642i)6-s + (0.173 + 0.300i)7-s + (0.500 − 0.866i)8-s + (0.407 − 0.342i)9-s + (2.97 − 5.14i)11-s + (0.939 + 1.62i)12-s + (3.09 + 1.12i)13-s + (0.0603 − 0.342i)14-s + (−0.939 + 0.342i)16-s + (4.23 + 3.55i)17-s − 0.532·18-s + (−4.11 + 1.43i)19-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (1.01 − 0.371i)3-s + (0.0868 + 0.492i)4-s + (−0.720 − 0.262i)6-s + (0.0656 + 0.113i)7-s + (0.176 − 0.306i)8-s + (0.135 − 0.114i)9-s + (0.896 − 1.55i)11-s + (0.271 + 0.469i)12-s + (0.857 + 0.312i)13-s + (0.0161 − 0.0914i)14-s + (−0.234 + 0.0855i)16-s + (1.02 + 0.862i)17-s − 0.125·18-s + (−0.944 + 0.328i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.500 + 0.865i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.500 + 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58863 - 0.916510i\)
\(L(\frac12)\) \(\approx\) \(1.58863 - 0.916510i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 \)
19 \( 1 + (4.11 - 1.43i)T \)
good3 \( 1 + (-1.76 + 0.642i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (-0.173 - 0.300i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.97 + 5.14i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.09 - 1.12i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-4.23 - 3.55i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (0.492 + 2.79i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-5.46 + 4.58i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (1.52 + 2.63i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.82T + 37T^{2} \)
41 \( 1 + (-6.41 + 2.33i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.0923 + 0.524i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (3.56 - 2.99i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-0.539 - 3.05i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (2.10 + 1.76i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (2.14 + 12.1i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-3.86 + 3.24i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (2.26 - 12.8i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-2.06 + 0.752i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-0.939 + 0.342i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-4.67 - 8.10i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (10.6 + 3.87i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-4.38 - 3.67i)T + (16.8 + 95.5i)T^{2} \)
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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

−9.741999176362771136852022437995, −8.817219367296743256951251986333, −8.387272192475174080508494452331, −7.87737646586310934327430894789, −6.51134241385654290397783439065, −5.85607216127850866186688339230, −4.03684670597464169344544427173, −3.38589884543918304546029370766, −2.28141745164990811502721704948, −1.10257065928358943307792487722, 1.38616817106571852617957574524, 2.74370124041999680862272381307, 3.88040862690190676346678927217, 4.81496306487019031816109546918, 6.06981862976679885353385966363, 7.02051825507127622723123230048, 7.75088215969255748492419599059, 8.644752823187264695827995615855, 9.254074069118330161408310585231, 9.863227758084797366697386413277

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