Properties

Label 2-950-95.54-c1-0-22
Degree $2$
Conductor $950$
Sign $-0.220 + 0.975i$
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  + (0.642 − 0.766i)2-s + (0.197 + 0.541i)3-s + (−0.173 − 0.984i)4-s + (0.541 + 0.197i)6-s + (−4.21 + 2.43i)7-s + (−0.866 − 0.500i)8-s + (2.04 − 1.71i)9-s + (2.68 − 4.64i)11-s + (0.499 − 0.288i)12-s + (1.31 − 3.62i)13-s + (−0.844 + 4.79i)14-s + (−0.939 + 0.342i)16-s + (−0.901 + 1.07i)17-s − 2.66i·18-s + (−4.35 + 0.226i)19-s + ⋯
L(s)  = 1  + (0.454 − 0.541i)2-s + (0.113 + 0.312i)3-s + (−0.0868 − 0.492i)4-s + (0.221 + 0.0804i)6-s + (−1.59 + 0.919i)7-s + (−0.306 − 0.176i)8-s + (0.681 − 0.571i)9-s + (0.809 − 1.40i)11-s + (0.144 − 0.0831i)12-s + (0.365 − 1.00i)13-s + (−0.225 + 1.28i)14-s + (−0.234 + 0.0855i)16-s + (−0.218 + 0.260i)17-s − 0.628i·18-s + (−0.998 + 0.0520i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.220 + 0.975i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.220 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01627 - 1.27107i\)
\(L(\frac12)\) \(\approx\) \(1.01627 - 1.27107i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.642 + 0.766i)T \)
5 \( 1 \)
19 \( 1 + (4.35 - 0.226i)T \)
good3 \( 1 + (-0.197 - 0.541i)T + (-2.29 + 1.92i)T^{2} \)
7 \( 1 + (4.21 - 2.43i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.68 + 4.64i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.31 + 3.62i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (0.901 - 1.07i)T + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (-5.25 + 0.927i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (-2.78 + 2.33i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (4.10 + 7.10i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 10.4iT - 37T^{2} \)
41 \( 1 + (-1.79 + 0.652i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.45 + 0.256i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (2.00 + 2.39i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (1.77 - 0.312i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (-5.61 - 4.70i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.40 - 7.99i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-5.42 - 6.46i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-1.04 + 5.93i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (0.436 + 1.19i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (15.6 - 5.68i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (12.5 - 7.23i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-9.02 - 3.28i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-3.90 + 4.65i)T + (-16.8 - 95.5i)T^{2} \)
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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

−9.824208516484406873926573921317, −9.052876962056785780112533945577, −8.593372039467422387170417540816, −6.93770048184712711262790321667, −6.10443514723948076584744571644, −5.66347784727236320522171569852, −4.04349252263207127252931648328, −3.46757061603514767368362057970, −2.56372957626234648666070117652, −0.66590929801901476156977444848, 1.62290097430566302531048837958, 3.15135856312215187362190734773, 4.20205750233015012993442712930, 4.78859001756310627088420355514, 6.49763734804341439754061068977, 6.81962117407090547978913111416, 7.25450765861012316944578301480, 8.586328340875631925431988384481, 9.486451326712664110961777611012, 10.08999451277529621316644671415

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