Properties

Label 2-950-19.6-c1-0-23
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  + (0.766 − 0.642i)2-s + (1.26 + 0.460i)3-s + (0.173 − 0.984i)4-s + (1.26 − 0.460i)6-s + (1.43 − 2.49i)7-s + (−0.500 − 0.866i)8-s + (−0.907 − 0.761i)9-s + (0.5 + 0.866i)11-s + (0.673 − 1.16i)12-s + (0.5 − 0.181i)13-s + (−0.500 − 2.83i)14-s + (−0.939 − 0.342i)16-s + (1.26 − 1.06i)17-s − 1.18·18-s + (3.79 − 2.15i)19-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (0.730 + 0.266i)3-s + (0.0868 − 0.492i)4-s + (0.516 − 0.188i)6-s + (0.544 − 0.942i)7-s + (−0.176 − 0.306i)8-s + (−0.302 − 0.253i)9-s + (0.150 + 0.261i)11-s + (0.194 − 0.336i)12-s + (0.138 − 0.0504i)13-s + (−0.133 − 0.757i)14-s + (−0.234 − 0.0855i)16-s + (0.307 − 0.257i)17-s − 0.279·18-s + (0.869 − 0.493i)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: $\chi_{950} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.320 + 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.25179 - 1.61480i\)
\(L(\frac12)\) \(\approx\) \(2.25179 - 1.61480i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.79 + 2.15i)T \)
good3 \( 1 + (-1.26 - 0.460i)T + (2.29 + 1.92i)T^{2} \)
7 \( 1 + (-1.43 + 2.49i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.181i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-1.26 + 1.06i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (0.194 - 1.10i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (1.81 + 1.52i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-0.847 + 1.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.59T + 37T^{2} \)
41 \( 1 + (4.85 + 1.76i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.88 - 10.6i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-1.39 - 1.16i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (2.08 - 11.8i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-6.72 + 5.64i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-1.08 + 6.15i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-1.93 - 1.62i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (1.02 + 5.79i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-4.68 - 1.70i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-8.05 - 2.93i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (5.32 - 9.22i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.694 - 0.252i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (8.88 - 7.45i)T + (16.8 - 95.5i)T^{2} \)
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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

−9.748747873644460520208775182025, −9.340499684211932504041789203246, −8.156048795037560563526338378074, −7.46692139575242708796787737526, −6.39570300728915969287789123280, −5.27414225436957034825701011275, −4.31546429257068465726390181989, −3.53477729072207506457770332839, −2.56549309293035955227356864265, −1.09593256446608352525394974499, 1.84007323661342107435031624584, 2.88240640880334423766979253980, 3.83259936320929575497840574195, 5.22878621451135242859693604857, 5.66933075072515077638488772705, 6.86148830728735699344081253287, 7.78995882703487576802250501185, 8.461702348427371327175540546593, 8.970847976258954161969585121050, 10.12033376613814373024054849004

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