Properties

Label 2-684-19.11-c1-0-3
Degree $2$
Conductor $684$
Sign $0.980 + 0.194i$
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  + 5·7-s + (−2.5 − 4.33i)13-s + (4 + 1.73i)19-s + (2.5 + 4.33i)25-s − 7·31-s + 11·37-s + (6.5 − 11.2i)43-s + 18·49-s + (0.5 + 0.866i)61-s + (−5.5 − 9.52i)67-s + (−8.5 + 14.7i)73-s + (−8.5 + 14.7i)79-s + (−12.5 − 21.6i)91-s + (−7 + 12.1i)97-s − 7·103-s + ⋯
L(s)  = 1  + 1.88·7-s + (−0.693 − 1.20i)13-s + (0.917 + 0.397i)19-s + (0.5 + 0.866i)25-s − 1.25·31-s + 1.80·37-s + (0.991 − 1.71i)43-s + 2.57·49-s + (0.0640 + 0.110i)61-s + (−0.671 − 1.16i)67-s + (−0.994 + 1.72i)73-s + (−0.956 + 1.65i)79-s + (−1.31 − 2.26i)91-s + (−0.710 + 1.23i)97-s − 0.689·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.194i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.980 + 0.194i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ 0.980 + 0.194i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81158 - 0.177909i\)
\(L(\frac12)\) \(\approx\) \(1.81158 - 0.177909i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + (-4 - 1.73i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 5T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 - 11T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.5 + 11.2i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (8.5 - 14.7i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.5 - 14.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7 - 12.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.64629209834504164122068357952, −9.597681772198807899677412653505, −8.607142089093781397251303625625, −7.73529321176240727748756940645, −7.32098649877668305079184523340, −5.58545930878387939553856047688, −5.17408386736887837476315613948, −4.01417680841473764695521307784, −2.57980856647541561766782447009, −1.24114364036567459768051279029, 1.40988922715689211484239282205, 2.56756070212563042614493454616, 4.33426593469222670368317764669, 4.81684582862015619294128793232, 5.94290793125298037350247414372, 7.25140502158361814858796396255, 7.79072969924423516048558821470, 8.800032022198911717833949569617, 9.531899008666477722016168516955, 10.68474007339944962749448773433

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