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Properties

Label 2-684-19.12-c2-0-14
Degree $2$
Conductor $684$
Sign $-0.429 + 0.902i$
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  + (2.19 + 3.80i)5-s + 1.44·7-s − 15.1·11-s + (−14.8 − 8.57i)13-s + (−9.77 − 16.9i)17-s + (−3.34 − 18.7i)19-s + (2.19 − 3.80i)23-s + (2.84 − 4.93i)25-s + (29.3 + 16.9i)29-s − 5.62i·31-s + (3.18 + 5.51i)35-s − 23.7i·37-s + (32.2 − 18.6i)41-s + (−15.9 − 27.6i)43-s + (−39.0 + 67.7i)47-s + ⋯
L(s)  = 1  + (0.439 + 0.760i)5-s + 0.207·7-s − 1.37·11-s + (−1.14 − 0.659i)13-s + (−0.574 − 0.995i)17-s + (−0.176 − 0.984i)19-s + (0.0955 − 0.165i)23-s + (0.113 − 0.197i)25-s + (1.01 + 0.583i)29-s − 0.181i·31-s + (0.0909 + 0.157i)35-s − 0.641i·37-s + (0.787 − 0.454i)41-s + (−0.371 − 0.643i)43-s + (−0.831 + 1.44i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.429 + 0.902i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.429 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.429 + 0.902i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ -0.429 + 0.902i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7856714316\)
\(L(\frac12)\) \(\approx\) \(0.7856714316\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + (3.34 + 18.7i)T \)
good5 \( 1 + (-2.19 - 3.80i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 1.44T + 49T^{2} \)
11 \( 1 + 15.1T + 121T^{2} \)
13 \( 1 + (14.8 + 8.57i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (9.77 + 16.9i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-2.19 + 3.80i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-29.3 - 16.9i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 5.62iT - 961T^{2} \)
37 \( 1 + 23.7iT - 1.36e3T^{2} \)
41 \( 1 + (-32.2 + 18.6i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (15.9 + 27.6i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (39.0 - 67.7i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-9.55 - 5.51i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-78.4 + 45.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-57.1 + 98.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (98.3 + 56.7i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (87.9 - 50.7i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-10.0 - 17.4i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (45.8 - 26.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 29.2T + 6.88e3T^{2} \)
89 \( 1 + (126. + 73.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-27 + 15.5i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.14810546877935423782821594828, −9.272339720028452320396711806878, −8.152542862964103083344921668410, −7.31520956056582018157708389315, −6.58155358448382155553931567134, −5.31373069875138037310819141455, −4.70660896809465377779865730358, −2.88648179682267841225486944030, −2.43164709314542113442842171321, −0.26182243302771700753875927416, 1.58042429262620762102350260251, 2.69155936118757774017153476117, 4.29911801644164864028920385924, 5.07174143078094635642333679946, 5.94417466759543871805794636902, 7.09490578288472517132994039005, 8.118414624906917512068599658231, 8.700380294693476108106879904677, 9.892122453358532102324304240457, 10.27574407539795921881092111414

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