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Properties

Label 2-684-171.103-c2-0-35
Degree $2$
Conductor $684$
Sign $-0.812 + 0.583i$
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  + (1.85 − 2.35i)3-s + (3.56 + 6.16i)5-s + (−5.30 − 9.19i)7-s + (−2.09 − 8.75i)9-s + (−9.75 − 16.8i)11-s + 1.06i·13-s + (21.1 + 3.07i)15-s + (−14.2 + 24.7i)17-s + (15.2 − 11.3i)19-s + (−31.5 − 4.58i)21-s − 31.1·23-s + (−12.8 + 22.2i)25-s + (−24.5 − 11.3i)27-s + (−37.8 − 21.8i)29-s + (22.3 + 12.8i)31-s + ⋯
L(s)  = 1  + (0.619 − 0.784i)3-s + (0.712 + 1.23i)5-s + (−0.758 − 1.31i)7-s + (−0.232 − 0.972i)9-s + (−0.887 − 1.53i)11-s + 0.0815i·13-s + (1.40 + 0.205i)15-s + (−0.840 + 1.45i)17-s + (0.802 − 0.596i)19-s + (−1.50 − 0.218i)21-s − 1.35·23-s + (−0.514 + 0.890i)25-s + (−0.907 − 0.420i)27-s + (−1.30 − 0.754i)29-s + (0.720 + 0.415i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.812 + 0.583i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ -0.812 + 0.583i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.371337877\)
\(L(\frac12)\) \(\approx\) \(1.371337877\)
\(L(2)\) 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 \( 1 + (-1.85 + 2.35i)T \)
19 \( 1 + (-15.2 + 11.3i)T \)
good5 \( 1 + (-3.56 - 6.16i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (5.30 + 9.19i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (9.75 + 16.8i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 1.06iT - 169T^{2} \)
17 \( 1 + (14.2 - 24.7i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + 31.1T + 529T^{2} \)
29 \( 1 + (37.8 + 21.8i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-22.3 - 12.8i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 33.5iT - 1.36e3T^{2} \)
41 \( 1 + (-23.5 + 13.5i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 - 32.3T + 1.84e3T^{2} \)
47 \( 1 + (17.5 - 30.4i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (2.36 - 1.36i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-33.0 + 19.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-17.8 + 30.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + 104. iT - 4.48e3T^{2} \)
71 \( 1 + (75.4 + 43.5i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-6.07 + 10.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 - 24.2iT - 6.24e3T^{2} \)
83 \( 1 + (38.3 + 66.3i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-110. + 63.7i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 83.3iT - 9.40e3T^{2} \)
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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

−10.06507894790004913388615846622, −9.086519234452575502413465780710, −7.949926814505647409657126166352, −7.33524173765363570717577895959, −6.28211034590701032780329940131, −6.01775273085095773395119035534, −3.86394131590349410917487414689, −3.14767920990218914750036221536, −2.10935055578963817362074890021, −0.40854944980694077712679677280, 2.01360507618127479350195449989, 2.76025002952763039010300990485, 4.33931661969313892013968181203, 5.19617840493372206091665074518, 5.72799782457755949879257390926, 7.28180721591419840975349163760, 8.324566636854773191291108637487, 9.136292792556072918278010926751, 9.693666542045529964458648359705, 10.03886825092129847434933403509

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