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Properties

Label 2-684-4.3-c2-0-14
Degree $2$
Conductor $684$
Sign $-0.965 + 0.258i$
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  + (1.58 + 1.21i)2-s + (1.03 + 3.86i)4-s − 1.46·5-s + 6.38i·7-s + (−3.06 + 7.39i)8-s + (−2.32 − 1.78i)10-s − 9.33i·11-s − 22.7·13-s + (−7.77 + 10.1i)14-s + (−13.8 + 8.00i)16-s + 5.17·17-s + 4.35i·19-s + (−1.52 − 5.67i)20-s + (11.3 − 14.8i)22-s + 37.0i·23-s + ⋯
L(s)  = 1  + (0.793 + 0.608i)2-s + (0.258 + 0.965i)4-s − 0.293·5-s + 0.912i·7-s + (−0.382 + 0.923i)8-s + (−0.232 − 0.178i)10-s − 0.849i·11-s − 1.74·13-s + (−0.555 + 0.724i)14-s + (−0.866 + 0.500i)16-s + 0.304·17-s + 0.229i·19-s + (−0.0760 − 0.283i)20-s + (0.516 − 0.673i)22-s + 1.61i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.965 + 0.258i$
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: \((2,\ 684,\ (\ :1),\ -0.965 + 0.258i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.284862927\)
\(L(\frac12)\) \(\approx\) \(1.284862927\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.58 - 1.21i)T \)
3 \( 1 \)
19 \( 1 - 4.35iT \)
good5 \( 1 + 1.46T + 25T^{2} \)
7 \( 1 - 6.38iT - 49T^{2} \)
11 \( 1 + 9.33iT - 121T^{2} \)
13 \( 1 + 22.7T + 169T^{2} \)
17 \( 1 - 5.17T + 289T^{2} \)
23 \( 1 - 37.0iT - 529T^{2} \)
29 \( 1 + 23.0T + 841T^{2} \)
31 \( 1 + 19.9iT - 961T^{2} \)
37 \( 1 - 24.3T + 1.36e3T^{2} \)
41 \( 1 - 57.3T + 1.68e3T^{2} \)
43 \( 1 + 78.8iT - 1.84e3T^{2} \)
47 \( 1 - 50.2iT - 2.20e3T^{2} \)
53 \( 1 + 82.4T + 2.80e3T^{2} \)
59 \( 1 - 86.4iT - 3.48e3T^{2} \)
61 \( 1 + 114.T + 3.72e3T^{2} \)
67 \( 1 - 64.7iT - 4.48e3T^{2} \)
71 \( 1 + 25.1iT - 5.04e3T^{2} \)
73 \( 1 + 58.1T + 5.32e3T^{2} \)
79 \( 1 - 47.6iT - 6.24e3T^{2} \)
83 \( 1 - 51.5iT - 6.88e3T^{2} \)
89 \( 1 + 4.37T + 7.92e3T^{2} \)
97 \( 1 - 16.1T + 9.40e3T^{2} \)
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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

−11.07023117588822653505045944483, −9.651873167140332935153430626932, −8.953643589773579220471039313061, −7.71486864305507290139426447563, −7.46067644378494103371680120409, −5.92924425133276875151372631901, −5.56482027131936608490034884917, −4.40618874819247040160200044675, −3.28974722474987458269701343126, −2.26654930217766025634622303377, 0.32182551158298891553853124470, 1.96161202540565155615521471194, 3.08909253546949800048266615692, 4.43189927135947574742011344785, 4.73463448839271896231329205595, 6.14046129539264637845072281595, 7.15543288464858484163576257235, 7.75773355946988513852273162727, 9.395012370914132609769575815258, 9.957792842893130731226515384488

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