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Properties

Label 2-1650-165.164-c1-0-61
Degree $2$
Conductor $1650$
Sign $-0.800 + 0.598i$
Analytic cond. $13.1753$
Root an. cond. $3.62978$
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  − i·2-s + (1.03 − 1.38i)3-s − 4-s + (−1.38 − 1.03i)6-s + 0.394·7-s + i·8-s + (−0.847 − 2.87i)9-s + (3.26 − 0.584i)11-s + (−1.03 + 1.38i)12-s + 2.37·13-s − 0.394i·14-s + 16-s − 0.906i·17-s + (−2.87 + 0.847i)18-s − 3.69i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.598 − 0.800i)3-s − 0.5·4-s + (−0.566 − 0.423i)6-s + 0.148·7-s + 0.353i·8-s + (−0.282 − 0.959i)9-s + (0.984 − 0.176i)11-s + (−0.299 + 0.400i)12-s + 0.660·13-s − 0.105i·14-s + 0.250·16-s − 0.219i·17-s + (−0.678 + 0.199i)18-s − 0.847i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 + 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.800 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1650\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 11\)
Sign: $-0.800 + 0.598i$
Analytic conductor: \(13.1753\)
Root analytic conductor: \(3.62978\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1650} (1649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1650,\ (\ :1/2),\ -0.800 + 0.598i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.056927581\)
\(L(\frac12)\) \(\approx\) \(2.056927581\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-1.03 + 1.38i)T \)
5 \( 1 \)
11 \( 1 + (-3.26 + 0.584i)T \)
good7 \( 1 - 0.394T + 7T^{2} \)
13 \( 1 - 2.37T + 13T^{2} \)
17 \( 1 + 0.906iT - 17T^{2} \)
19 \( 1 + 3.69iT - 19T^{2} \)
23 \( 1 + 3.54T + 23T^{2} \)
29 \( 1 - 3.37T + 29T^{2} \)
31 \( 1 - 3.66T + 31T^{2} \)
37 \( 1 + 4.58iT - 37T^{2} \)
41 \( 1 - 8.14T + 41T^{2} \)
43 \( 1 - 2.60T + 43T^{2} \)
47 \( 1 + 3.39T + 47T^{2} \)
53 \( 1 + 6.45T + 53T^{2} \)
59 \( 1 - 7.76iT - 59T^{2} \)
61 \( 1 + 12.8iT - 61T^{2} \)
67 \( 1 + 11.9iT - 67T^{2} \)
71 \( 1 + 2.58iT - 71T^{2} \)
73 \( 1 + 11.8T + 73T^{2} \)
79 \( 1 + 2.56iT - 79T^{2} \)
83 \( 1 - 1.69iT - 83T^{2} \)
89 \( 1 - 15.3iT - 89T^{2} \)
97 \( 1 + 10.1iT - 97T^{2} \)
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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

−9.128670516011751685804575648671, −8.357892646436144219168594548857, −7.65337682535684144363831500729, −6.60531620594539569249070979757, −6.01063360160669799488454049497, −4.65131192433164714883185137642, −3.74006015236988721022689573711, −2.86818276742126678783771842118, −1.82182435348730054007158473177, −0.793482758599529785619531680051, 1.53858614625961826600836821076, 3.03766195127233239759620578189, 4.03764309541146021828920527831, 4.53848128498747076499566256621, 5.72661851154128928536851473072, 6.36432912090745553003598892175, 7.44119988982237410263284117349, 8.295989477195855550597403891785, 8.689332254301655999799041068215, 9.673827594660648382559338850851

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