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Properties

Label 2-684-76.7-c0-0-2
Degree $2$
Conductor $684$
Sign $0.977 + 0.211i$
Analytic cond. $0.341360$
Root an. cond. $0.584260$
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  + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.707 − 1.22i)5-s + i·7-s + (0.707 − 0.707i)8-s + (1.36 − 0.366i)10-s − 1.41i·11-s + (0.5 − 0.866i)13-s + (−0.965 − 0.258i)14-s + (0.500 + 0.866i)16-s − i·19-s + 1.41i·20-s + (1.36 + 0.366i)22-s + (1.22 + 0.707i)23-s + (−0.499 + 0.866i)25-s + (0.707 + 0.707i)26-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.707 − 1.22i)5-s + i·7-s + (0.707 − 0.707i)8-s + (1.36 − 0.366i)10-s − 1.41i·11-s + (0.5 − 0.866i)13-s + (−0.965 − 0.258i)14-s + (0.500 + 0.866i)16-s − i·19-s + 1.41i·20-s + (1.36 + 0.366i)22-s + (1.22 + 0.707i)23-s + (−0.499 + 0.866i)25-s + (0.707 + 0.707i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.977 + 0.211i$
Analytic conductor: \(0.341360\)
Root analytic conductor: \(0.584260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :0),\ 0.977 + 0.211i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6591954139\)
\(L(\frac12)\) \(\approx\) \(0.6591954139\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 \)
19 \( 1 + iT \)
good5 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 - iT - T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + iT - T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−10.65432586193119046990098132776, −9.125310331037272911583546338928, −8.919544049587971227193325298360, −8.205071330055155422648068467838, −7.35989083935623609126236799932, −5.95467059176782995886409746803, −5.46673929003997776242664054341, −4.51204795151145310712983397807, −3.22219826610681404897451657560, −0.884902620780147459825508399959, 1.73012079626924733978588375793, 3.14317193113038057989371441228, 3.96614323793529465882074025846, 4.78459348626309175912940417740, 6.69674962109647417504543317936, 7.24269445120049559464084354034, 8.098863740985984264460307630708, 9.250207101813000458688940765732, 10.16554637694789441914139892782, 10.72252415070891778283177392164

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