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Properties

Label 2-684-4.3-c2-0-6
Degree $2$
Conductor $684$
Sign $-0.904 - 0.427i$
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  + (−1.07 − 1.68i)2-s + (−1.70 + 3.61i)4-s − 6.66·5-s + 12.1i·7-s + (7.93 − 0.984i)8-s + (7.13 + 11.2i)10-s − 4.93i·11-s + 11.2·13-s + (20.5 − 13.0i)14-s + (−10.1 − 12.3i)16-s + 17.2·17-s + 4.35i·19-s + (11.3 − 24.1i)20-s + (−8.34 + 5.28i)22-s + 16.0i·23-s + ⋯
L(s)  = 1  + (−0.535 − 0.844i)2-s + (−0.427 + 0.904i)4-s − 1.33·5-s + 1.73i·7-s + (0.992 − 0.123i)8-s + (0.713 + 1.12i)10-s − 0.448i·11-s + 0.865·13-s + (1.46 − 0.928i)14-s + (−0.635 − 0.772i)16-s + 1.01·17-s + 0.229i·19-s + (0.569 − 1.20i)20-s + (−0.379 + 0.240i)22-s + 0.697i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.427i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.904 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.904 - 0.427i$
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: \((2,\ 684,\ (\ :1),\ -0.904 - 0.427i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1803810380\)
\(L(\frac12)\) \(\approx\) \(0.1803810380\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.07 + 1.68i)T \)
3 \( 1 \)
19 \( 1 - 4.35iT \)
good5 \( 1 + 6.66T + 25T^{2} \)
7 \( 1 - 12.1iT - 49T^{2} \)
11 \( 1 + 4.93iT - 121T^{2} \)
13 \( 1 - 11.2T + 169T^{2} \)
17 \( 1 - 17.2T + 289T^{2} \)
23 \( 1 - 16.0iT - 529T^{2} \)
29 \( 1 + 50.3T + 841T^{2} \)
31 \( 1 + 6.35iT - 961T^{2} \)
37 \( 1 + 36.0T + 1.36e3T^{2} \)
41 \( 1 - 11.8T + 1.68e3T^{2} \)
43 \( 1 - 59.2iT - 1.84e3T^{2} \)
47 \( 1 - 35.3iT - 2.20e3T^{2} \)
53 \( 1 + 38.6T + 2.80e3T^{2} \)
59 \( 1 + 89.8iT - 3.48e3T^{2} \)
61 \( 1 + 85.8T + 3.72e3T^{2} \)
67 \( 1 + 103. iT - 4.48e3T^{2} \)
71 \( 1 + 67.2iT - 5.04e3T^{2} \)
73 \( 1 - 109.T + 5.32e3T^{2} \)
79 \( 1 + 66.3iT - 6.24e3T^{2} \)
83 \( 1 - 4.01iT - 6.88e3T^{2} \)
89 \( 1 + 53.9T + 7.92e3T^{2} \)
97 \( 1 + 62.8T + 9.40e3T^{2} \)
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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

−11.09908549038927580622027688326, −9.582215883079758594350426796334, −9.022775981970302270460290439813, −8.074800241289922065194902113257, −7.73607921734186597231462563904, −6.14051631337897025505576478483, −5.07891386777769235299913731919, −3.69485264599545268389600003257, −3.11822918566060942668381836553, −1.64313048312908289390471610441, 0.090309615450276730781867975010, 1.22617277718299553893319104486, 3.71782111244963128543561745794, 4.20423346360283247327887437967, 5.42403010215735023740413205782, 6.79704214480314231023445092620, 7.34921009955219789331484908917, 7.930003443938255305199392861421, 8.799093794179802499178259998817, 9.948416990261159363309582796322

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