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Properties

Label 2-536-536.307-c0-0-0
Degree $2$
Conductor $536$
Sign $-0.0348 + 0.999i$
Analytic cond. $0.267498$
Root an. cond. $0.517202$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.928 − 0.371i)2-s + (−0.759 − 0.876i)3-s + (0.723 − 0.690i)4-s + (−1.03 − 0.531i)6-s + (0.415 − 0.909i)8-s + (−0.0492 + 0.342i)9-s + (−0.738 + 0.380i)11-s + (−1.15 − 0.110i)12-s + (0.0475 − 0.998i)16-s + (0.341 + 0.325i)17-s + (0.0815 + 0.336i)18-s + (−0.0748 + 0.0588i)19-s + (−0.544 + 0.627i)22-s + (−1.11 + 0.326i)24-s + (0.415 + 0.909i)25-s + ⋯
L(s)  = 1  + (0.928 − 0.371i)2-s + (−0.759 − 0.876i)3-s + (0.723 − 0.690i)4-s + (−1.03 − 0.531i)6-s + (0.415 − 0.909i)8-s + (−0.0492 + 0.342i)9-s + (−0.738 + 0.380i)11-s + (−1.15 − 0.110i)12-s + (0.0475 − 0.998i)16-s + (0.341 + 0.325i)17-s + (0.0815 + 0.336i)18-s + (−0.0748 + 0.0588i)19-s + (−0.544 + 0.627i)22-s + (−1.11 + 0.326i)24-s + (0.415 + 0.909i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0348 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0348 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(536\)    =    \(2^{3} \cdot 67\)
Sign: $-0.0348 + 0.999i$
Analytic conductor: \(0.267498\)
Root analytic conductor: \(0.517202\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{536} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 536,\ (\ :0),\ -0.0348 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.178716867\)
\(L(\frac12)\) \(\approx\) \(1.178716867\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.928 + 0.371i)T \)
67 \( 1 + (-0.841 - 0.540i)T \)
good3 \( 1 + (0.759 + 0.876i)T + (-0.142 + 0.989i)T^{2} \)
5 \( 1 + (-0.415 - 0.909i)T^{2} \)
7 \( 1 + (-0.235 - 0.971i)T^{2} \)
11 \( 1 + (0.738 - 0.380i)T + (0.580 - 0.814i)T^{2} \)
13 \( 1 + (0.327 + 0.945i)T^{2} \)
17 \( 1 + (-0.341 - 0.325i)T + (0.0475 + 0.998i)T^{2} \)
19 \( 1 + (0.0748 - 0.0588i)T + (0.235 - 0.971i)T^{2} \)
23 \( 1 + (0.786 + 0.618i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.327 - 0.945i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.0671 - 0.276i)T + (-0.888 - 0.458i)T^{2} \)
43 \( 1 + (-1.70 + 0.500i)T + (0.841 - 0.540i)T^{2} \)
47 \( 1 + (-0.928 + 0.371i)T^{2} \)
53 \( 1 + (-0.841 - 0.540i)T^{2} \)
59 \( 1 + (0.827 - 1.81i)T + (-0.654 - 0.755i)T^{2} \)
61 \( 1 + (-0.580 - 0.814i)T^{2} \)
71 \( 1 + (-0.0475 + 0.998i)T^{2} \)
73 \( 1 + (1.28 + 0.663i)T + (0.580 + 0.814i)T^{2} \)
79 \( 1 + (-0.981 - 0.189i)T^{2} \)
83 \( 1 + (-0.0800 + 1.68i)T + (-0.995 - 0.0950i)T^{2} \)
89 \( 1 + (1.21 - 1.40i)T + (-0.142 - 0.989i)T^{2} \)
97 \( 1 + (0.723 + 1.25i)T + (-0.5 + 0.866i)T^{2} \)
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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

−11.02613502357486987637318271751, −10.35893815653419513077585527109, −9.228484905577083057917337561726, −7.67123798118601420232188653272, −7.04070877125344826750435500064, −6.00674530016372628018800763409, −5.38521575439452501596958529538, −4.19261520436264544633650064226, −2.80311667849474606731546959897, −1.41551709662210133830461713591, 2.59319488173354515801319615532, 3.87334550023884068552633106899, 4.83860026865063916150838642069, 5.50422499790549751392157420781, 6.38277949641098427665802251669, 7.53908752440575771375117946614, 8.433762321073903150896051950671, 9.758164405754283399666663979696, 10.69682907616438465072554114010, 11.18733317735588926204891120729

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