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Properties

Label 2-950-19.16-c1-0-25
Degree $2$
Conductor $950$
Sign $0.995 + 0.0923i$
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 + (2.56 − 0.934i)3-s + (0.173 + 0.984i)4-s + (2.56 + 0.934i)6-s + (−1.06 − 1.85i)7-s + (−0.500 + 0.866i)8-s + (3.42 − 2.87i)9-s + (0.926 − 1.60i)11-s + (1.36 + 2.36i)12-s + (5.18 + 1.88i)13-s + (0.371 − 2.10i)14-s + (−0.939 + 0.342i)16-s + (−2.82 − 2.37i)17-s + 4.46·18-s + (1.21 − 4.18i)19-s + ⋯
L(s)  = 1  + (0.541 + 0.454i)2-s + (1.48 − 0.539i)3-s + (0.0868 + 0.492i)4-s + (1.04 + 0.381i)6-s + (−0.403 − 0.699i)7-s + (−0.176 + 0.306i)8-s + (1.14 − 0.957i)9-s + (0.279 − 0.483i)11-s + (0.394 + 0.683i)12-s + (1.43 + 0.522i)13-s + (0.0991 − 0.562i)14-s + (−0.234 + 0.0855i)16-s + (−0.686 − 0.575i)17-s + 1.05·18-s + (0.278 − 0.960i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.995 + 0.0923i$
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: \((2,\ 950,\ (\ :1/2),\ 0.995 + 0.0923i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.46181 - 0.160199i\)
\(L(\frac12)\) \(\approx\) \(3.46181 - 0.160199i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.21 + 4.18i)T \)
good3 \( 1 + (-2.56 + 0.934i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (1.06 + 1.85i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.926 + 1.60i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.18 - 1.88i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (2.82 + 2.37i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-0.511 - 2.89i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (5.36 - 4.50i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-2.12 - 3.68i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.49T + 37T^{2} \)
41 \( 1 + (-1.47 + 0.536i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.404 - 2.29i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (8.59 - 7.20i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (1.11 + 6.32i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-10.8 - 9.07i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (0.0461 + 0.261i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (11.3 - 9.49i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.722 + 4.09i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (14.4 - 5.27i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-2.70 + 0.985i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (7.00 + 12.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.86 + 0.676i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (8.28 + 6.94i)T + (16.8 + 95.5i)T^{2} \)
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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

−9.676705754609692590127340035458, −8.864182560525697901449307707955, −8.457439910232350806115780244933, −7.26394604189421997933614342810, −6.95020355148844685851931320903, −5.90804016720166280047045055818, −4.45220030217824079015446354911, −3.55876205819939517288320790828, −2.89052744354691158308786821871, −1.41494681721868050583155135855, 1.79289528796439461586639713652, 2.76114976500928857244885096901, 3.69550432733136379169808521052, 4.25881728911162378653820779545, 5.66661524204205838706398333869, 6.45587701227230694927714559997, 7.86679391750735476276434202531, 8.524568129031966154419514741064, 9.283810189898106513553818221723, 9.920397814244747838878290625843

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