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Properties

Label 2-290-145.34-c1-0-10
Degree $2$
Conductor $290$
Sign $0.0124 + 0.999i$
Analytic cond. $2.31566$
Root an. cond. $1.52172$
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  + (−0.623 − 0.781i)2-s + (−0.489 − 2.14i)3-s + (−0.222 + 0.974i)4-s + (0.704 + 2.12i)5-s + (−1.37 + 1.71i)6-s + (4.41 − 1.00i)7-s + (0.900 − 0.433i)8-s + (−1.65 + 0.796i)9-s + (1.22 − 1.87i)10-s + (1.70 − 3.53i)11-s + 2.19·12-s + (−1.21 + 2.52i)13-s + (−3.54 − 2.82i)14-s + (4.20 − 2.54i)15-s + (−0.900 − 0.433i)16-s − 1.61·17-s + ⋯
L(s)  = 1  + (−0.440 − 0.552i)2-s + (−0.282 − 1.23i)3-s + (−0.111 + 0.487i)4-s + (0.314 + 0.949i)5-s + (−0.559 + 0.701i)6-s + (1.66 − 0.380i)7-s + (0.318 − 0.153i)8-s + (−0.551 + 0.265i)9-s + (0.385 − 0.592i)10-s + (0.513 − 1.06i)11-s + 0.634·12-s + (−0.336 + 0.699i)13-s + (−0.946 − 0.754i)14-s + (1.08 − 0.657i)15-s + (−0.225 − 0.108i)16-s − 0.391·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0124 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0124 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(290\)    =    \(2 \cdot 5 \cdot 29\)
Sign: $0.0124 + 0.999i$
Analytic conductor: \(2.31566\)
Root analytic conductor: \(1.52172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{290} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 290,\ (\ :1/2),\ 0.0124 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.823118 - 0.812951i\)
\(L(\frac12)\) \(\approx\) \(0.823118 - 0.812951i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 + (-0.704 - 2.12i)T \)
29 \( 1 + (-1.19 + 5.25i)T \)
good3 \( 1 + (0.489 + 2.14i)T + (-2.70 + 1.30i)T^{2} \)
7 \( 1 + (-4.41 + 1.00i)T + (6.30 - 3.03i)T^{2} \)
11 \( 1 + (-1.70 + 3.53i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (1.21 - 2.52i)T + (-8.10 - 10.1i)T^{2} \)
17 \( 1 + 1.61T + 17T^{2} \)
19 \( 1 + (-2.96 - 0.677i)T + (17.1 + 8.24i)T^{2} \)
23 \( 1 + (2.46 + 1.96i)T + (5.11 + 22.4i)T^{2} \)
31 \( 1 + (5.89 - 4.69i)T + (6.89 - 30.2i)T^{2} \)
37 \( 1 + (-7.83 + 3.77i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 - 1.60iT - 41T^{2} \)
43 \( 1 + (-0.637 + 0.799i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (-1.18 - 0.572i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (6.05 - 4.83i)T + (11.7 - 51.6i)T^{2} \)
59 \( 1 + 3.30T + 59T^{2} \)
61 \( 1 + (-11.9 + 2.72i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 + (-6.20 - 12.8i)T + (-41.7 + 52.3i)T^{2} \)
71 \( 1 + (0.960 + 0.462i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (9.42 - 11.8i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + (3.68 + 7.65i)T + (-49.2 + 61.7i)T^{2} \)
83 \( 1 + (3.39 + 0.773i)T + (74.7 + 36.0i)T^{2} \)
89 \( 1 + (11.0 - 8.83i)T + (19.8 - 86.7i)T^{2} \)
97 \( 1 + (3.66 - 16.0i)T + (-87.3 - 42.0i)T^{2} \)
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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

−11.27519802704701739208006478897, −11.17072465100434080996560388579, −9.814118438919320806756890145963, −8.514658073211420604195524117918, −7.64881720295075440882626460816, −6.91376478019784828149972154328, −5.79078592393963267657779454410, −4.11311667196119753338318876329, −2.35979067804432340996149948056, −1.28369741211561202730479542326, 1.71473390510610925362803289610, 4.32019187699002600533689817548, 4.96943020629602822815722973122, 5.64743097477939286704136017740, 7.43621498109887914540956535977, 8.336275935088352003767518696230, 9.320897652018033280164422411089, 9.861489817110037818951949177138, 10.97278972726967975273693083386, 11.76433898959136319119711291804

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