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Properties

Label 2-684-19.10-c0-0-0
Degree $2$
Conductor $684$
Sign $0.877 + 0.479i$
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  + (0.766 − 1.32i)7-s + (−0.439 + 0.524i)13-s + (0.5 − 0.866i)19-s + (−0.766 − 0.642i)25-s + (1.70 + 0.984i)31-s + 1.96i·37-s + (−0.326 − 0.118i)43-s + (−0.673 − 1.16i)49-s + (−1.76 + 0.642i)61-s + (−1.26 − 0.223i)67-s + (−1.43 + 1.20i)73-s + (−0.826 − 0.984i)79-s + (0.358 + 0.984i)91-s + (1.70 − 0.300i)97-s + (−0.592 + 0.342i)103-s + ⋯
L(s)  = 1  + (0.766 − 1.32i)7-s + (−0.439 + 0.524i)13-s + (0.5 − 0.866i)19-s + (−0.766 − 0.642i)25-s + (1.70 + 0.984i)31-s + 1.96i·37-s + (−0.326 − 0.118i)43-s + (−0.673 − 1.16i)49-s + (−1.76 + 0.642i)61-s + (−1.26 − 0.223i)67-s + (−1.43 + 1.20i)73-s + (−0.826 − 0.984i)79-s + (0.358 + 0.984i)91-s + (1.70 − 0.300i)97-s + (−0.592 + 0.342i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.877 + 0.479i$
Analytic conductor: \(0.341360\)
Root analytic conductor: \(0.584260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :0),\ 0.877 + 0.479i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9968512470\)
\(L(\frac12)\) \(\approx\) \(0.9968512470\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (0.766 + 0.642i)T^{2} \)
7 \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.439 - 0.524i)T + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (-0.939 + 0.342i)T^{2} \)
23 \( 1 + (0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.939 + 0.342i)T^{2} \)
31 \( 1 + (-1.70 - 0.984i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 - 1.96iT - T^{2} \)
41 \( 1 + (-0.173 + 0.984i)T^{2} \)
43 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (-0.939 - 0.342i)T^{2} \)
53 \( 1 + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (0.939 - 0.342i)T^{2} \)
61 \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (1.26 + 0.223i)T + (0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
79 \( 1 + (0.826 + 0.984i)T + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.173 - 0.984i)T^{2} \)
97 \( 1 + (-1.70 + 0.300i)T + (0.939 - 0.342i)T^{2} \)
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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

−10.46499480529182779628486583871, −10.00713258364811846927613241155, −8.839683260643161565239585774548, −7.921534652886123090905863234367, −7.18371184896453927356848030637, −6.33814988689385048340169448147, −4.84971570336997752323416601007, −4.36050194069171985722726175557, −2.96198235580469654820454909790, −1.35416418438351362985570696626, 1.86105366631770519492106893107, 2.98382476335954801826486347200, 4.40567542085640903675844548099, 5.48820105473896310955081072819, 6.02146560119169986206488824293, 7.51211568925926207715297801157, 8.107382254557670565263733760911, 9.049693579606438219113646086144, 9.808516859409187009698174340247, 10.80232997388172990291275207437

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