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Properties

Label 2-50-5.2-c4-0-0
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: $\chi_{50} (7, \cdot )$
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

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

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