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Properties

Label 2-50-5.3-c4-0-5
Degree $2$
Conductor $50$
Sign $-0.973 - 0.229i$
Analytic cond. $5.16849$
Root an. cond. $2.27343$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 − 2i)2-s + (−9 − 9i)3-s − 8i·4-s − 36·6-s + (−29 + 29i)7-s + (−16 − 16i)8-s + 81i·9-s − 118·11-s + (−72 + 72i)12-s + (−69 − 69i)13-s + 116i·14-s − 64·16-s + (271 − 271i)17-s + (162 + 162i)18-s − 280i·19-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−1 − i)3-s − 0.5i·4-s − 6-s + (−0.591 + 0.591i)7-s + (−0.250 − 0.250i)8-s + i·9-s − 0.975·11-s + (−0.5 + 0.5i)12-s + (−0.408 − 0.408i)13-s + 0.591i·14-s − 0.250·16-s + (0.937 − 0.937i)17-s + (0.5 + 0.5i)18-s − 0.775i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $-0.973 - 0.229i$
Analytic conductor: \(5.16849\)
Root analytic conductor: \(2.27343\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :2),\ -0.973 - 0.229i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0877587 + 0.753721i\)
\(L(\frac12)\) \(\approx\) \(0.0877587 + 0.753721i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 + 2i)T \)
5 \( 1 \)
good3 \( 1 + (9 + 9i)T + 81iT^{2} \)
7 \( 1 + (29 - 29i)T - 2.40e3iT^{2} \)
11 \( 1 + 118T + 1.46e4T^{2} \)
13 \( 1 + (69 + 69i)T + 2.85e4iT^{2} \)
17 \( 1 + (-271 + 271i)T - 8.35e4iT^{2} \)
19 \( 1 + 280iT - 1.30e5T^{2} \)
23 \( 1 + (269 + 269i)T + 2.79e5iT^{2} \)
29 \( 1 + 680iT - 7.07e5T^{2} \)
31 \( 1 - 202T + 9.23e5T^{2} \)
37 \( 1 + (-651 + 651i)T - 1.87e6iT^{2} \)
41 \( 1 - 1.68e3T + 2.82e6T^{2} \)
43 \( 1 + (1.08e3 + 1.08e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (1.26e3 - 1.26e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (-611 - 611i)T + 7.89e6iT^{2} \)
59 \( 1 + 1.16e3iT - 1.21e7T^{2} \)
61 \( 1 + 5.59e3T + 1.38e7T^{2} \)
67 \( 1 + (-751 + 751i)T - 2.01e7iT^{2} \)
71 \( 1 - 6.44e3T + 2.54e7T^{2} \)
73 \( 1 + (-2.95e3 - 2.95e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 1.05e4iT - 3.89e7T^{2} \)
83 \( 1 + (-6.23e3 - 6.23e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 1.44e4iT - 6.27e7T^{2} \)
97 \( 1 + (-7.31e3 + 7.31e3i)T - 8.85e7iT^{2} \)
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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

−13.75659294600431034796105602860, −12.71346444490093053490863374752, −12.14191744793533284270707247701, −11.01367151529108703032981729047, −9.663982871311451058281074713992, −7.62344010833921794109869465659, −6.21084730516439869112954004768, −5.17076854249274166905869460748, −2.63813215005198984489545750794, −0.42754194622724493681379385876, 3.71305015523928919960072780484, 5.07994717672298426671836276290, 6.20778940332173655034141268111, 7.81838317986705921462637489712, 9.811978040297879966635525788444, 10.61952159184843810480449755343, 11.98225094237428945169719951543, 13.11448581170423635391398609710, 14.48515281833405262838710004047, 15.62206721800081959092510969411

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