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Properties

Label 2-684-76.75-c1-0-33
Degree $2$
Conductor $684$
Sign $-0.508 + 0.860i$
Analytic cond. $5.46176$
Root an. cond. $2.33704$
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  + (−0.331 − 1.37i)2-s + (−1.78 + 0.910i)4-s + 2.56·5-s − 4.15i·7-s + (1.84 + 2.14i)8-s + (−0.848 − 3.52i)10-s − 2.33i·11-s + 4.29i·13-s + (−5.70 + 1.37i)14-s + (2.34 − 3.24i)16-s + 17-s + (3.68 − 2.33i)19-s + (−4.56 + 2.33i)20-s + (−3.20 + 0.772i)22-s − 1.82i·23-s + ⋯
L(s)  = 1  + (−0.234 − 0.972i)2-s + (−0.890 + 0.455i)4-s + 1.14·5-s − 1.56i·7-s + (0.650 + 0.759i)8-s + (−0.268 − 1.11i)10-s − 0.703i·11-s + 1.19i·13-s + (−1.52 + 0.367i)14-s + (0.585 − 0.810i)16-s + 0.242·17-s + (0.844 − 0.535i)19-s + (−1.01 + 0.521i)20-s + (−0.683 + 0.164i)22-s − 0.379i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.508 + 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.508 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.508 + 0.860i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ -0.508 + 0.860i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.717431 - 1.25721i\)
\(L(\frac12)\) \(\approx\) \(0.717431 - 1.25721i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.331 + 1.37i)T \)
3 \( 1 \)
19 \( 1 + (-3.68 + 2.33i)T \)
good5 \( 1 - 2.56T + 5T^{2} \)
7 \( 1 + 4.15iT - 7T^{2} \)
11 \( 1 + 2.33iT - 11T^{2} \)
13 \( 1 - 4.29iT - 13T^{2} \)
17 \( 1 - T + 17T^{2} \)
23 \( 1 + 1.82iT - 23T^{2} \)
29 \( 1 + 1.20iT - 29T^{2} \)
31 \( 1 + 1.32T + 31T^{2} \)
37 \( 1 + 5.49iT - 37T^{2} \)
41 \( 1 + 5.49iT - 41T^{2} \)
43 \( 1 + 1.30iT - 43T^{2} \)
47 \( 1 - 6.99iT - 47T^{2} \)
53 \( 1 + 9.79iT - 53T^{2} \)
59 \( 1 + 6.33T + 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 + 0.290T + 67T^{2} \)
71 \( 1 + 2.06T + 71T^{2} \)
73 \( 1 + 0.123T + 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 - 11.9iT - 83T^{2} \)
89 \( 1 - 14.0iT - 89T^{2} \)
97 \( 1 + 2.41iT - 97T^{2} \)
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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

−10.17699965031652983574477699549, −9.571002786256613693565126475608, −8.814534150737270287523611893401, −7.63490613627609564529377407993, −6.74537089363087041447674388978, −5.50644277730167573720856588433, −4.39410838841864430559277541072, −3.48209661634644784295589221296, −2.11353887722777141780056188225, −0.908831982967593977060327674384, 1.66764227789094988240662601416, 3.05930762286822218011386955855, 4.87487040504711450624698745432, 5.67189222110130331891905049728, 6.01323342791582327004583415956, 7.27570975892626225388114518426, 8.209751506085669649133458697130, 9.023403856616896832675544952940, 9.795488593394451608346001816088, 10.22720628429029217864983323628

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