Properties

Label 2-684-4.3-c2-0-10
Degree $2$
Conductor $684$
Sign $-0.978 + 0.203i$
Analytic cond. $18.6376$
Root an. cond. $4.31713$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.55 + 1.26i)2-s + (0.815 + 3.91i)4-s − 2.61·5-s − 10.9i·7-s + (−3.67 + 7.10i)8-s + (−4.05 − 3.29i)10-s + 21.8i·11-s − 4.95·13-s + (13.8 − 17.0i)14-s + (−14.6 + 6.38i)16-s − 20.1·17-s + 4.35i·19-s + (−2.13 − 10.2i)20-s + (−27.5 + 33.9i)22-s − 9.15i·23-s + ⋯
L(s)  = 1  + (0.775 + 0.630i)2-s + (0.203 + 0.978i)4-s − 0.522·5-s − 1.57i·7-s + (−0.459 + 0.888i)8-s + (−0.405 − 0.329i)10-s + 1.98i·11-s − 0.381·13-s + (0.990 − 1.21i)14-s + (−0.916 + 0.399i)16-s − 1.18·17-s + 0.229i·19-s + (−0.106 − 0.511i)20-s + (−1.25 + 1.54i)22-s − 0.398i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.203i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.978 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.978 + 0.203i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ -0.978 + 0.203i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9210847138\)
\(L(\frac12)\) \(\approx\) \(0.9210847138\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.55 - 1.26i)T \)
3 \( 1 \)
19 \( 1 - 4.35iT \)
good5 \( 1 + 2.61T + 25T^{2} \)
7 \( 1 + 10.9iT - 49T^{2} \)
11 \( 1 - 21.8iT - 121T^{2} \)
13 \( 1 + 4.95T + 169T^{2} \)
17 \( 1 + 20.1T + 289T^{2} \)
23 \( 1 + 9.15iT - 529T^{2} \)
29 \( 1 + 1.24T + 841T^{2} \)
31 \( 1 - 39.9iT - 961T^{2} \)
37 \( 1 + 56.6T + 1.36e3T^{2} \)
41 \( 1 + 31.6T + 1.68e3T^{2} \)
43 \( 1 + 23.3iT - 1.84e3T^{2} \)
47 \( 1 - 64.7iT - 2.20e3T^{2} \)
53 \( 1 - 80.8T + 2.80e3T^{2} \)
59 \( 1 + 12.2iT - 3.48e3T^{2} \)
61 \( 1 - 64.3T + 3.72e3T^{2} \)
67 \( 1 + 82.9iT - 4.48e3T^{2} \)
71 \( 1 + 16.4iT - 5.04e3T^{2} \)
73 \( 1 + 63.6T + 5.32e3T^{2} \)
79 \( 1 + 19.2iT - 6.24e3T^{2} \)
83 \( 1 + 29.3iT - 6.88e3T^{2} \)
89 \( 1 + 21.4T + 7.92e3T^{2} \)
97 \( 1 - 74.7T + 9.40e3T^{2} \)
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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

−10.71812841990172569721970864648, −10.03148507510032427033523729370, −8.806201134740277951748417404718, −7.68809723480896924483537642854, −7.14241347577473009174347535179, −6.63670094786929602142268157689, −4.97559902842683861347743093090, −4.39430650901955520427786855510, −3.62760578733177185604801637724, −2.02162818321676907018965966874, 0.23238386788116055232568005388, 2.11559388817479317435766233279, 3.06306415776007352863277800659, 4.04407994678327005945213593587, 5.39174484690835141468047761798, 5.82811078510175699636707545129, 6.89528323967679613894552774657, 8.437896250087098664502914694273, 8.866954213755877067428821723629, 9.939528895420697314900696333021

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