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Properties

Label 2-950-95.24-c1-0-11
Degree $2$
Conductor $950$
Sign $0.320 - 0.947i$
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 + (−0.634 − 0.111i)3-s + (−0.766 + 0.642i)4-s + (−0.111 − 0.634i)6-s + (−0.369 + 0.213i)7-s + (−0.866 − 0.500i)8-s + (−2.42 − 0.883i)9-s + (1.50 − 2.60i)11-s + (0.558 − 0.322i)12-s + (4.71 − 0.831i)13-s + (−0.327 − 0.274i)14-s + (0.173 − 0.984i)16-s + (1.62 + 4.46i)17-s − 2.58i·18-s + (2.34 + 3.67i)19-s + ⋯
L(s)  = 1  + (0.241 + 0.664i)2-s + (−0.366 − 0.0646i)3-s + (−0.383 + 0.321i)4-s + (−0.0457 − 0.259i)6-s + (−0.139 + 0.0807i)7-s + (−0.306 − 0.176i)8-s + (−0.809 − 0.294i)9-s + (0.453 − 0.786i)11-s + (0.161 − 0.0930i)12-s + (1.30 − 0.230i)13-s + (−0.0874 − 0.0733i)14-s + (0.0434 − 0.246i)16-s + (0.394 + 1.08i)17-s − 0.609i·18-s + (0.537 + 0.843i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.320 - 0.947i$
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: \((2,\ 950,\ (\ :1/2),\ 0.320 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19720 + 0.858902i\)
\(L(\frac12)\) \(\approx\) \(1.19720 + 0.858902i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.34 - 3.67i)T \)
good3 \( 1 + (0.634 + 0.111i)T + (2.81 + 1.02i)T^{2} \)
7 \( 1 + (0.369 - 0.213i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.50 + 2.60i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.71 + 0.831i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-1.62 - 4.46i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (-3.36 - 4.00i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (0.536 + 0.195i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-0.113 - 0.196i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.76iT - 37T^{2} \)
41 \( 1 + (-1.60 + 9.12i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (0.652 - 0.777i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (-2.46 + 6.78i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (-6.95 - 8.29i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (-0.330 + 0.120i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.0978 + 0.0820i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (0.0895 - 0.246i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (-0.343 - 0.287i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-8.41 - 1.48i)T + (68.5 + 24.9i)T^{2} \)
79 \( 1 + (-2.31 + 13.1i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (0.954 - 0.550i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.75 - 9.98i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (5.28 + 14.5i)T + (-74.3 + 62.3i)T^{2} \)
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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

−10.25484461483747926465684989490, −9.041774682171104490052794152082, −8.540968235228014222934096808380, −7.71126668095353420084185427137, −6.49500902520247982646957274584, −5.93670000039412148922874576801, −5.35394072747980042754365516343, −3.81717368435733899647953204379, −3.24399774968779460708485006079, −1.15409502620800615676823586193, 0.848052496174206494385191604120, 2.40110658753532435080596208967, 3.43140176045598143790407621661, 4.56456652302426443507128412141, 5.34711301744624715742111774804, 6.33368126431299008653816604744, 7.22103840014594093779882524264, 8.446489536310321341745413131418, 9.195015530238992216298942393221, 9.914966888798340093560264006686

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