Properties

Label 2-950-95.24-c1-0-19
Degree $2$
Conductor $950$
Sign $0.967 + 0.252i$
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  + (−0.342 − 0.939i)2-s + (3.13 + 0.552i)3-s + (−0.766 + 0.642i)4-s + (−0.552 − 3.13i)6-s + (2.89 − 1.67i)7-s + (0.866 + 0.500i)8-s + (6.68 + 2.43i)9-s + (−3.09 + 5.35i)11-s + (−2.75 + 1.58i)12-s + (0.727 − 0.128i)13-s + (−2.56 − 2.15i)14-s + (0.173 − 0.984i)16-s + (0.296 + 0.815i)17-s − 7.11i·18-s + (2.59 + 3.49i)19-s + ⋯
L(s)  = 1  + (−0.241 − 0.664i)2-s + (1.80 + 0.318i)3-s + (−0.383 + 0.321i)4-s + (−0.225 − 1.27i)6-s + (1.09 − 0.632i)7-s + (0.306 + 0.176i)8-s + (2.22 + 0.810i)9-s + (−0.932 + 1.61i)11-s + (−0.794 + 0.458i)12-s + (0.201 − 0.0355i)13-s + (−0.685 − 0.574i)14-s + (0.0434 − 0.246i)16-s + (0.0720 + 0.197i)17-s − 1.67i·18-s + (0.596 + 0.802i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.967 + 0.252i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.967 + 0.252i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.72575 - 0.349114i\)
\(L(\frac12)\) \(\approx\) \(2.72575 - 0.349114i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.342 + 0.939i)T \)
5 \( 1 \)
19 \( 1 + (-2.59 - 3.49i)T \)
good3 \( 1 + (-3.13 - 0.552i)T + (2.81 + 1.02i)T^{2} \)
7 \( 1 + (-2.89 + 1.67i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.09 - 5.35i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.727 + 0.128i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.296 - 0.815i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (0.735 + 0.875i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (7.32 + 2.66i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (3.23 + 5.60i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.68iT - 37T^{2} \)
41 \( 1 + (-1.68 + 9.54i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (1.19 - 1.42i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (0.113 - 0.311i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (4.56 + 5.44i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (-4.56 + 1.66i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-5.53 + 4.64i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (1.00 - 2.74i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (5.27 + 4.42i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-7.69 - 1.35i)T + (68.5 + 24.9i)T^{2} \)
79 \( 1 + (-0.399 + 2.26i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (9.91 - 5.72i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.404 + 2.29i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (2.34 + 6.43i)T + (-74.3 + 62.3i)T^{2} \)
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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

−9.923365156419902646951285841860, −9.316450966217128566414370644168, −8.311404150298120906823536104468, −7.66459131970121268486068040861, −7.36503980378026163825095312806, −5.20931338118365132634278428024, −4.25317731970654580045300610289, −3.65582702827183872828118666946, −2.28107845525465051817311212708, −1.75445878591089901178803550006, 1.39460706801176973221522771827, 2.66409235612953206126245143707, 3.50846778686065564798997345382, 4.88759866600493345780784147961, 5.76490034439821565243721955042, 7.09161133276319047056984290375, 7.80716602609762830887510039342, 8.417916657082621402151866014032, 8.844906394558581266507986263112, 9.611464620950414448825850805481

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