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Properties

Label 2-684-171.7-c1-0-18
Degree $2$
Conductor $684$
Sign $-0.0293 + 0.999i$
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  + (1.24 − 1.20i)3-s + (0.745 − 1.29i)5-s + (0.903 − 1.56i)7-s + (0.118 − 2.99i)9-s + (0.400 − 0.694i)11-s − 1.50·13-s + (−0.618 − 2.50i)15-s + (−1.44 − 2.49i)17-s + (−2.63 + 3.47i)19-s + (−0.750 − 3.03i)21-s + 3.01·23-s + (1.38 + 2.40i)25-s + (−3.45 − 3.88i)27-s + (2.82 + 4.90i)29-s + (−1.97 − 3.42i)31-s + ⋯
L(s)  = 1  + (0.720 − 0.693i)3-s + (0.333 − 0.577i)5-s + (0.341 − 0.591i)7-s + (0.0393 − 0.999i)9-s + (0.120 − 0.209i)11-s − 0.417·13-s + (−0.159 − 0.647i)15-s + (−0.349 − 0.605i)17-s + (−0.604 + 0.796i)19-s + (−0.163 − 0.663i)21-s + 0.628·23-s + (0.277 + 0.481i)25-s + (−0.664 − 0.747i)27-s + (0.525 + 0.910i)29-s + (−0.355 − 0.615i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0293 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0293 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.0293 + 0.999i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ -0.0293 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38468 - 1.42589i\)
\(L(\frac12)\) \(\approx\) \(1.38468 - 1.42589i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.24 + 1.20i)T \)
19 \( 1 + (2.63 - 3.47i)T \)
good5 \( 1 + (-0.745 + 1.29i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.903 + 1.56i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.400 + 0.694i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.50T + 13T^{2} \)
17 \( 1 + (1.44 + 2.49i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 - 3.01T + 23T^{2} \)
29 \( 1 + (-2.82 - 4.90i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.97 + 3.42i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.08T + 37T^{2} \)
41 \( 1 + (-3.66 + 6.34i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 3.66T + 43T^{2} \)
47 \( 1 + (-0.0411 - 0.0713i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.16 + 2.01i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.33 + 5.77i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.37 - 9.30i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + 8.50T + 67T^{2} \)
71 \( 1 + (-0.0790 - 0.137i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.67 - 2.90i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 0.856T + 79T^{2} \)
83 \( 1 + (-2.39 + 4.14i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.35 + 5.81i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.87T + 97T^{2} \)
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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

−10.16407302489341106753224674405, −9.128702202830717354350372937041, −8.653079368143620153429605663685, −7.57600007498947330436382075691, −6.97902052577699233872274818780, −5.82477860231082146533381460313, −4.68908899265001632305841826382, −3.56191742412280979346506743250, −2.24450485225542157221479626857, −1.02706791393802880121905623657, 2.12290458415548938373368175537, 2.90571939855669024191230578125, 4.23753806858574832865633441348, 5.07916681459664336239593465381, 6.26737585747560475182150461166, 7.25418327049087348554300565966, 8.347488402839598385948240022576, 8.953186451711221311668151385719, 9.838951739428297861799588222811, 10.59793847860724954152661078716

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