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Properties

Label 2-1620-60.47-c0-0-1
Degree $2$
Conductor $1620$
Sign $0.685 - 0.727i$
Analytic cond. $0.808485$
Root an. cond. $0.899158$
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  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.258 + 0.965i)5-s + (0.707 + 0.707i)8-s + (−0.866 − 0.500i)10-s + (1.36 − 1.36i)13-s − 1.00·16-s + (1.22 − 1.22i)17-s + (0.965 − 0.258i)20-s + (−0.866 + 0.499i)25-s + 1.93i·26-s − 0.517·29-s + (0.707 − 0.707i)32-s + 1.73i·34-s + (−0.366 − 0.366i)37-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.258 + 0.965i)5-s + (0.707 + 0.707i)8-s + (−0.866 − 0.500i)10-s + (1.36 − 1.36i)13-s − 1.00·16-s + (1.22 − 1.22i)17-s + (0.965 − 0.258i)20-s + (−0.866 + 0.499i)25-s + 1.93i·26-s − 0.517·29-s + (0.707 − 0.707i)32-s + 1.73i·34-s + (−0.366 − 0.366i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.685 - 0.727i$
Analytic conductor: \(0.808485\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ 0.685 - 0.727i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8901107498\)
\(L(\frac12)\) \(\approx\) \(0.8901107498\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 + (-0.258 - 0.965i)T \)
good7 \( 1 - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
17 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + 0.517T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.73T + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + 1.93T + T^{2} \)
97 \( 1 + (-1 - i)T + iT^{2} \)
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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

−9.770217252845421468517550955307, −8.840480723934652112318360795607, −7.916818154610366941406776341843, −7.46113393532626488792660839961, −6.50277407895115028335081722984, −5.81327548134124492508212084478, −5.16099427089876877106686403881, −3.61891125075105146221780404912, −2.68459359348795849812777953675, −1.15854273303398251252109637878, 1.25624921452704045376982932348, 1.99855369129168967017989672494, 3.63076968828590932984229228565, 4.09691269347198747243017800293, 5.36147662680864194794653012052, 6.29995325091486605152598353206, 7.28410715489761301418363976320, 8.456787260494644439196895848035, 8.547457932999345023261401870472, 9.496248295291187349088759935573

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