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Properties

Label 2-950-19.6-c1-0-12
Degree $2$
Conductor $950$
Sign $-0.153 - 0.988i$
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 + (2.74 + 0.998i)3-s + (0.173 − 0.984i)4-s + (−2.74 + 0.998i)6-s + (−0.366 + 0.635i)7-s + (0.500 + 0.866i)8-s + (4.23 + 3.54i)9-s + (2.78 + 4.81i)11-s + (1.45 − 2.52i)12-s + (−2.81 + 1.02i)13-s + (−0.127 − 0.722i)14-s + (−0.939 − 0.342i)16-s + (−1.79 + 1.50i)17-s − 5.52·18-s + (4.10 − 1.46i)19-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (1.58 + 0.576i)3-s + (0.0868 − 0.492i)4-s + (−1.11 + 0.407i)6-s + (−0.138 + 0.240i)7-s + (0.176 + 0.306i)8-s + (1.41 + 1.18i)9-s + (0.838 + 1.45i)11-s + (0.421 − 0.729i)12-s + (−0.779 + 0.283i)13-s + (−0.0340 − 0.193i)14-s + (−0.234 − 0.0855i)16-s + (−0.436 + 0.365i)17-s − 1.30·18-s + (0.941 − 0.336i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.153 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.153 - 0.988i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.153 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33504 + 1.55774i\)
\(L(\frac12)\) \(\approx\) \(1.33504 + 1.55774i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.10 + 1.46i)T \)
good3 \( 1 + (-2.74 - 0.998i)T + (2.29 + 1.92i)T^{2} \)
7 \( 1 + (0.366 - 0.635i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.78 - 4.81i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.81 - 1.02i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (1.79 - 1.50i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-0.521 + 2.95i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (6.50 + 5.45i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-0.667 + 1.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.27T + 37T^{2} \)
41 \( 1 + (-6.72 - 2.44i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.0151 + 0.0858i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-8.49 - 7.12i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (1.91 - 10.8i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (0.740 - 0.620i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-1.76 + 9.99i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-0.0888 - 0.0745i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (0.207 + 1.17i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (11.3 + 4.13i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (1.74 + 0.634i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-8.09 + 14.0i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-15.2 + 5.55i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (3.04 - 2.55i)T + (16.8 - 95.5i)T^{2} \)
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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

−9.776816153014478793890642351690, −9.293588118610731019189227137106, −8.920560028159640380661815794023, −7.60155103636592799259126751232, −7.42762744163527694872023293551, −6.18724389246078879696017962883, −4.71672829655409357431855411140, −4.09673627722122037535551687536, −2.71904672194027816480478259695, −1.84635434530509633284418318101, 1.00866700506574891814248421738, 2.22837726869816982262423090874, 3.29700922253472864789894985730, 3.77188898916397432817329939382, 5.50732543811560370362197239799, 6.94347501251873207684633969487, 7.41557937776296488835202579694, 8.335514012526762148708040094133, 8.998115282132380683958883470493, 9.467304367867466271390185499284

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