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Properties

Label 2-290-145.109-c1-0-12
Degree $2$
Conductor $290$
Sign $-0.887 + 0.461i$
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  + (−0.222 − 0.974i)2-s + (−0.577 − 0.278i)3-s + (−0.900 + 0.433i)4-s + (−1.06 + 1.96i)5-s + (−0.142 + 0.625i)6-s + (1.73 − 3.60i)7-s + (0.623 + 0.781i)8-s + (−1.61 − 2.02i)9-s + (2.15 + 0.600i)10-s + (−0.710 − 0.566i)11-s + 0.641·12-s + (−4.16 − 3.32i)13-s + (−3.90 − 0.890i)14-s + (1.16 − 0.839i)15-s + (0.623 − 0.781i)16-s − 0.839·17-s + ⋯
L(s)  = 1  + (−0.157 − 0.689i)2-s + (−0.333 − 0.160i)3-s + (−0.450 + 0.216i)4-s + (−0.476 + 0.879i)5-s + (−0.0582 + 0.255i)6-s + (0.656 − 1.36i)7-s + (0.220 + 0.276i)8-s + (−0.538 − 0.674i)9-s + (0.681 + 0.189i)10-s + (−0.214 − 0.170i)11-s + 0.185·12-s + (−1.15 − 0.921i)13-s + (−1.04 − 0.238i)14-s + (0.300 − 0.216i)15-s + (0.155 − 0.195i)16-s − 0.203·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(290\)    =    \(2 \cdot 5 \cdot 29\)
Sign: $-0.887 + 0.461i$
Analytic conductor: \(2.31566\)
Root analytic conductor: \(1.52172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{290} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 290,\ (\ :1/2),\ -0.887 + 0.461i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.153694 - 0.628154i\)
\(L(\frac12)\) \(\approx\) \(0.153694 - 0.628154i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.222 + 0.974i)T \)
5 \( 1 + (1.06 - 1.96i)T \)
29 \( 1 + (-4.71 - 2.59i)T \)
good3 \( 1 + (0.577 + 0.278i)T + (1.87 + 2.34i)T^{2} \)
7 \( 1 + (-1.73 + 3.60i)T + (-4.36 - 5.47i)T^{2} \)
11 \( 1 + (0.710 + 0.566i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (4.16 + 3.32i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + 0.839T + 17T^{2} \)
19 \( 1 + (1.47 + 3.05i)T + (-11.8 + 14.8i)T^{2} \)
23 \( 1 + (0.675 + 0.154i)T + (20.7 + 9.97i)T^{2} \)
31 \( 1 + (-0.378 + 0.0863i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + (-0.732 - 0.918i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + 8.57iT - 41T^{2} \)
43 \( 1 + (1.61 - 7.08i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-4.26 + 5.35i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (5.56 - 1.27i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 - 13.8T + 59T^{2} \)
61 \( 1 + (-4.46 + 9.27i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 + (0.110 - 0.0877i)T + (14.9 - 65.3i)T^{2} \)
71 \( 1 + (1.67 - 2.10i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (1.01 - 4.45i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + (7.26 - 5.79i)T + (17.5 - 77.0i)T^{2} \)
83 \( 1 + (-4.98 - 10.3i)T + (-51.7 + 64.8i)T^{2} \)
89 \( 1 + (-3.98 + 0.910i)T + (80.1 - 38.6i)T^{2} \)
97 \( 1 + (-10.6 + 5.13i)T + (60.4 - 75.8i)T^{2} \)
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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

−11.26345277358708805402134549714, −10.64202136143490385248562175615, −9.923951458546242199303284073285, −8.453786522968863588330026907673, −7.51253212620424923870505379310, −6.69779727060218979057361931272, −5.08931520286035759225781116252, −3.87691365601716682812420154846, −2.71410743974975653619585995861, −0.51897880636746638546812580183, 2.19515833676134668108166345760, 4.51561064633830058295367074357, 5.12501852502993292232244624959, 6.07021999131075182640252569476, 7.58481206085459698474568049595, 8.386137492935632211308037975283, 9.049628908215871082381104779449, 10.15620419345315576955378450031, 11.60479701702542440843329778839, 11.97137049232661956126138974706

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