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Properties

Label 2-684-12.11-c1-0-12
Degree $2$
Conductor $684$
Sign $0.816 + 0.577i$
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  − 1.41i·2-s − 2.00·4-s − 1.41i·5-s + 4i·7-s + 2.82i·8-s − 2.00·10-s + 2.82·11-s + 5.65·14-s + 4.00·16-s + 1.41i·17-s + i·19-s + 2.82i·20-s − 4.00i·22-s + 8.48·23-s + 2.99·25-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.00·4-s − 0.632i·5-s + 1.51i·7-s + 1.00i·8-s − 0.632·10-s + 0.852·11-s + 1.51·14-s + 1.00·16-s + 0.342i·17-s + 0.229i·19-s + 0.632i·20-s − 0.852i·22-s + 1.76·23-s + 0.599·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ 0.816 + 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32923 - 0.422481i\)
\(L(\frac12)\) \(\approx\) \(1.32923 - 0.422481i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 \)
19 \( 1 - iT \)
good5 \( 1 + 1.41iT - 5T^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 - 2.82T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 1.41iT - 17T^{2} \)
23 \( 1 - 8.48T + 23T^{2} \)
29 \( 1 + 1.41iT - 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 - 7.07iT - 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + 7.07iT - 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + 5.65T + 71T^{2} \)
73 \( 1 - 16T + 73T^{2} \)
79 \( 1 - 8iT - 79T^{2} \)
83 \( 1 + 2.82T + 83T^{2} \)
89 \( 1 + 9.89iT - 89T^{2} \)
97 \( 1 - 12T + 97T^{2} \)
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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

−10.44810323777185191075361315872, −9.391427683183749550514699564355, −8.839699562551088148000987844030, −8.369561475998455504368364672351, −6.77344925375867812522250946736, −5.51454382438427818437017154306, −4.92106191344507177949297932178, −3.64505204188669406694029353962, −2.52571275027077279461558864784, −1.28598130497286603549240491997, 0.931311764816628739949317098991, 3.26419127663248636757753718939, 4.18100429364531822115853297235, 5.12920006800727218852271671712, 6.49560466910955150349541522088, 7.00673194436709450861538644197, 7.60317857903777940392125158317, 8.808874693869484430391375114200, 9.551293226260740100158771285937, 10.54377976372621378093846833032

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