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Properties

Label 2-950-25.6-c1-0-31
Degree $2$
Conductor $950$
Sign $0.267 + 0.963i$
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  + (0.309 + 0.951i)2-s + (−1.03 + 0.752i)3-s + (−0.809 + 0.587i)4-s + (−0.232 − 2.22i)5-s + (−1.03 − 0.752i)6-s − 0.203·7-s + (−0.809 − 0.587i)8-s + (−0.420 + 1.29i)9-s + (2.04 − 0.908i)10-s + (0.113 + 0.347i)11-s + (0.395 − 1.21i)12-s + (−0.621 + 1.91i)13-s + (−0.0628 − 0.193i)14-s + (1.91 + 2.12i)15-s + (0.309 − 0.951i)16-s + (−2.04 − 1.48i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.598 + 0.434i)3-s + (−0.404 + 0.293i)4-s + (−0.103 − 0.994i)5-s + (−0.423 − 0.307i)6-s − 0.0768·7-s + (−0.286 − 0.207i)8-s + (−0.140 + 0.431i)9-s + (0.646 − 0.287i)10-s + (0.0340 + 0.104i)11-s + (0.114 − 0.351i)12-s + (−0.172 + 0.530i)13-s + (−0.0167 − 0.0516i)14-s + (0.494 + 0.549i)15-s + (0.0772 − 0.237i)16-s + (−0.495 − 0.359i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.267 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.267 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.267 + 0.963i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.267 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.409177 - 0.311047i\)
\(L(\frac12)\) \(\approx\) \(0.409177 - 0.311047i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (0.232 + 2.22i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (1.03 - 0.752i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + 0.203T + 7T^{2} \)
11 \( 1 + (-0.113 - 0.347i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.621 - 1.91i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.04 + 1.48i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 + (1.68 + 5.17i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-5.25 + 3.81i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (5.38 + 3.91i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-3.56 + 10.9i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.00566 + 0.0174i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 5.31T + 43T^{2} \)
47 \( 1 + (-6.10 + 4.43i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.492 + 0.357i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.03 - 6.25i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.689 - 2.12i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (6.08 + 4.41i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (2.26 - 1.64i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.636 - 1.95i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.04 - 1.48i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (2.86 + 2.08i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-1.83 - 5.65i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-8.51 + 6.18i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.724694699688308306464794940863, −8.959651527471159773873148530176, −8.207787350316444006049909689889, −7.31762443959407241556209791243, −6.23604980693719576779546743039, −5.47791608039584015905494415023, −4.59771557275595796582137219186, −4.12002915918624587098029976609, −2.27609294705444434876362270441, −0.24155417696059465468546547438, 1.48967964724392883080667356827, 2.92474652571895521385679341283, 3.64615921789449615274985559922, 4.97585046342929766345372674075, 6.02585289117212102300103134522, 6.61075951873369465025969598544, 7.56670949289841621108907184538, 8.614242522353225953030041500037, 9.664631865705271246038176451243, 10.40532440115227947144952684711

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