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Properties

Label 2-950-25.11-c1-0-31
Degree $2$
Conductor $950$
Sign $0.268 + 0.963i$
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  + (0.809 − 0.587i)2-s + (0.140 − 0.432i)3-s + (0.309 − 0.951i)4-s + (0.187 − 2.22i)5-s + (−0.140 − 0.432i)6-s + 3.42·7-s + (−0.309 − 0.951i)8-s + (2.25 + 1.64i)9-s + (−1.15 − 1.91i)10-s + (0.563 − 0.409i)11-s + (−0.368 − 0.267i)12-s + (3.42 + 2.49i)13-s + (2.76 − 2.01i)14-s + (−0.937 − 0.394i)15-s + (−0.809 − 0.587i)16-s + (−0.0122 − 0.0378i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.0811 − 0.249i)3-s + (0.154 − 0.475i)4-s + (0.0839 − 0.996i)5-s + (−0.0574 − 0.176i)6-s + 1.29·7-s + (−0.109 − 0.336i)8-s + (0.753 + 0.547i)9-s + (−0.366 − 0.604i)10-s + (0.169 − 0.123i)11-s + (−0.106 − 0.0772i)12-s + (0.951 + 0.691i)13-s + (0.739 − 0.537i)14-s + (−0.242 − 0.101i)15-s + (−0.202 − 0.146i)16-s + (−0.00298 − 0.00917i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.268 + 0.963i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (761, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.268 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23050 - 1.69390i\)
\(L(\frac12)\) \(\approx\) \(2.23050 - 1.69390i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-0.187 + 2.22i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 + (-0.140 + 0.432i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 - 3.42T + 7T^{2} \)
11 \( 1 + (-0.563 + 0.409i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-3.42 - 2.49i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.0122 + 0.0378i)T + (-13.7 + 9.99i)T^{2} \)
23 \( 1 + (4.12 - 3.00i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.316 + 0.972i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.867 - 2.67i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.47 + 3.25i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-0.816 - 0.593i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 8.26T + 43T^{2} \)
47 \( 1 + (-1.96 + 6.04i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.459 + 1.41i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (10.6 + 7.76i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (12.2 - 8.87i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.43 - 7.49i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (0.845 - 2.60i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-3.29 + 2.39i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.836 + 2.57i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.954 + 2.93i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-9.26 + 6.73i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (0.388 - 1.19i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.993389106164628474423482956185, −8.947589696314823191835853225273, −8.246573270475805637797668685660, −7.44402069568143969892064428071, −6.24041509454864049784673547730, −5.24380409328078079082563369214, −4.55893343997418278651754372761, −3.76149374353156588984668912183, −1.88986744430478714872005362371, −1.40752822301567842931395647521, 1.67009143464027681667063902279, 3.08864476900829549431328547869, 4.01969935870520410680865246986, 4.86830460103271543843117719176, 6.00884910624356331572611919448, 6.68671863591333891391048872923, 7.67765514888848102479433364860, 8.264820624370620747417490185468, 9.378957847174123098750463691591, 10.44736298378128938856988447546

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