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Properties

Label 2-950-95.94-c2-0-18
Degree $2$
Conductor $950$
Sign $0.447 - 0.894i$
Analytic cond. $25.8856$
Root an. cond. $5.08779$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s − 2.82·3-s + 2.00·4-s − 4.00·6-s + 5i·7-s + 2.82·8-s − 0.999·9-s + 5·11-s − 5.65·12-s + 16.9·13-s + 7.07i·14-s + 4.00·16-s − 25i·17-s − 1.41·18-s − 19·19-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.942·3-s + 0.500·4-s − 0.666·6-s + 0.714i·7-s + 0.353·8-s − 0.111·9-s + 0.454·11-s − 0.471·12-s + 1.30·13-s + 0.505i·14-s + 0.250·16-s − 1.47i·17-s − 0.0785·18-s − 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(25.8856\)
Root analytic conductor: \(5.08779\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1),\ 0.447 - 0.894i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.938790962\)
\(L(\frac12)\) \(\approx\) \(1.938790962\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
5 \( 1 \)
19 \( 1 + 19T \)
good3 \( 1 + 2.82T + 9T^{2} \)
7 \( 1 - 5iT - 49T^{2} \)
11 \( 1 - 5T + 121T^{2} \)
13 \( 1 - 16.9T + 169T^{2} \)
17 \( 1 + 25iT - 289T^{2} \)
23 \( 1 - 10iT - 529T^{2} \)
29 \( 1 - 42.4iT - 841T^{2} \)
31 \( 1 - 42.4iT - 961T^{2} \)
37 \( 1 + 25.4T + 1.36e3T^{2} \)
41 \( 1 - 42.4iT - 1.68e3T^{2} \)
43 \( 1 + 5iT - 1.84e3T^{2} \)
47 \( 1 - 5iT - 2.20e3T^{2} \)
53 \( 1 - 25.4T + 2.80e3T^{2} \)
59 \( 1 - 84.8iT - 3.48e3T^{2} \)
61 \( 1 - 95T + 3.72e3T^{2} \)
67 \( 1 - 110.T + 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 25iT - 5.32e3T^{2} \)
79 \( 1 - 42.4iT - 6.24e3T^{2} \)
83 \( 1 - 130iT - 6.88e3T^{2} \)
89 \( 1 + 127. iT - 7.92e3T^{2} \)
97 \( 1 - 16.9T + 9.40e3T^{2} \)
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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

−10.30437408509979788222751167282, −9.015875846672298013206992844756, −8.474530464264558650249284385109, −6.99507993769180502154117182676, −6.44889325346574263907650603283, −5.51833275951812777069009477140, −5.01412959574632150875472440168, −3.76269218029041561894440953319, −2.69750261238785729839034498164, −1.18514118286561523305270457551, 0.62448460280903851943592870089, 2.05717609543744960218401816638, 3.77510834026642587319535815608, 4.18475222604996293682693281312, 5.48872659261189199449305083621, 6.23486182746920644326563576831, 6.65156020595452800335562417200, 7.988058105116050484644162132571, 8.704387494581755547240373256966, 10.09340080748154049776840094258

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