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Properties

Label 2-950-95.18-c1-0-29
Degree $2$
Conductor $950$
Sign $-0.869 + 0.493i$
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  + (−0.707 − 0.707i)2-s + (1.91 − 1.91i)3-s + 1.00i·4-s − 2.70·6-s + (2.95 − 2.95i)7-s + (0.707 − 0.707i)8-s − 4.33i·9-s − 3.90·11-s + (1.91 + 1.91i)12-s + (−4.27 + 4.27i)13-s − 4.17·14-s − 1.00·16-s + (3.82 − 3.82i)17-s + (−3.06 + 3.06i)18-s + (−0.797 − 4.28i)19-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (1.10 − 1.10i)3-s + 0.500i·4-s − 1.10·6-s + (1.11 − 1.11i)7-s + (0.250 − 0.250i)8-s − 1.44i·9-s − 1.17·11-s + (0.552 + 0.552i)12-s + (−1.18 + 1.18i)13-s − 1.11·14-s − 0.250·16-s + (0.926 − 0.926i)17-s + (−0.722 + 0.722i)18-s + (−0.183 − 0.983i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 + 0.493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.869 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.869 + 0.493i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.869 + 0.493i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.429093 - 1.62418i\)
\(L(\frac12)\) \(\approx\) \(0.429093 - 1.62418i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 \)
19 \( 1 + (0.797 + 4.28i)T \)
good3 \( 1 + (-1.91 + 1.91i)T - 3iT^{2} \)
7 \( 1 + (-2.95 + 2.95i)T - 7iT^{2} \)
11 \( 1 + 3.90T + 11T^{2} \)
13 \( 1 + (4.27 - 4.27i)T - 13iT^{2} \)
17 \( 1 + (-3.82 + 3.82i)T - 17iT^{2} \)
23 \( 1 + (4.44 + 4.44i)T + 23iT^{2} \)
29 \( 1 - 2.10T + 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + (1.05 + 1.05i)T + 37iT^{2} \)
41 \( 1 + 6.48iT - 41T^{2} \)
43 \( 1 + (-6.63 - 6.63i)T + 43iT^{2} \)
47 \( 1 + (-2.35 + 2.35i)T - 47iT^{2} \)
53 \( 1 + (-3.83 + 3.83i)T - 53iT^{2} \)
59 \( 1 - 14.9T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + (-4.04 - 4.04i)T + 67iT^{2} \)
71 \( 1 - 1.70iT - 71T^{2} \)
73 \( 1 + (-5.04 - 5.04i)T + 73iT^{2} \)
79 \( 1 + 11.8T + 79T^{2} \)
83 \( 1 + (-2.04 - 2.04i)T + 83iT^{2} \)
89 \( 1 + 10.9T + 89T^{2} \)
97 \( 1 + (3.35 + 3.35i)T + 97iT^{2} \)
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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

−9.677237886386881865784681093382, −8.626974890176265415280558677085, −8.055094886160500718763444994241, −7.19182733657622739296842077634, −7.08363457921400891370923794516, −5.07997423742429300790184115274, −4.14805585216876608036014831900, −2.69520199429538894804360258809, −2.10785736604473156676111740786, −0.798443537136384279332582010488, 2.04554034062433769721690706950, 2.93665948190078288962304657739, 4.23419006206416145916721139300, 5.40356463488813698993225973317, 5.62246795345854138248826528681, 7.69853845122633190890122036254, 8.009548049314795177344664091705, 8.511604003356664416162672501928, 9.598381227487571649187751555892, 10.12727128459354085242154000321

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