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Properties

Label 2-684-171.11-c2-0-0
Degree $2$
Conductor $684$
Sign $-0.985 + 0.170i$
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  + (−2.99 − 0.179i)3-s − 0.944i·5-s + (−5.01 + 8.69i)7-s + (8.93 + 1.07i)9-s + (2.50 + 1.44i)11-s + (3.63 − 6.30i)13-s + (−0.169 + 2.82i)15-s + (15.0 + 8.71i)17-s + (−18.3 + 4.74i)19-s + (16.5 − 25.1i)21-s + (−11.5 − 6.65i)23-s + 24.1·25-s + (−26.5 − 4.83i)27-s − 16.6i·29-s + (4.71 + 8.17i)31-s + ⋯
L(s)  = 1  + (−0.998 − 0.0599i)3-s − 0.188i·5-s + (−0.716 + 1.24i)7-s + (0.992 + 0.119i)9-s + (0.228 + 0.131i)11-s + (0.279 − 0.484i)13-s + (−0.0113 + 0.188i)15-s + (0.888 + 0.512i)17-s + (−0.968 + 0.249i)19-s + (0.790 − 1.19i)21-s + (−0.501 − 0.289i)23-s + 0.964·25-s + (−0.983 − 0.179i)27-s − 0.573i·29-s + (0.152 + 0.263i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.170i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.985 + 0.170i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ -0.985 + 0.170i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1076192342\)
\(L(\frac12)\) \(\approx\) \(0.1076192342\)
\(L(2)\) 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 \( 1 + (2.99 + 0.179i)T \)
19 \( 1 + (18.3 - 4.74i)T \)
good5 \( 1 + 0.944iT - 25T^{2} \)
7 \( 1 + (5.01 - 8.69i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-2.50 - 1.44i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-3.63 + 6.30i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-15.0 - 8.71i)T + (144.5 + 250. i)T^{2} \)
23 \( 1 + (11.5 + 6.65i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 16.6iT - 841T^{2} \)
31 \( 1 + (-4.71 - 8.17i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 63.1T + 1.36e3T^{2} \)
41 \( 1 - 34.1iT - 1.68e3T^{2} \)
43 \( 1 + (23.8 + 41.3i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + 42.6iT - 2.20e3T^{2} \)
53 \( 1 + (69.9 - 40.3i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 - 92.1iT - 3.48e3T^{2} \)
61 \( 1 + 56.9T + 3.72e3T^{2} \)
67 \( 1 + (39.6 - 68.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (61.9 + 35.7i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-63.6 + 110. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (3.36 + 5.82i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (101. + 58.7i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (87.6 - 50.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (42.5 + 73.6i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.58638861113405730389496374646, −10.09684738443768099334063576673, −9.009635100420624453281342159491, −8.243978795059510081791892450082, −6.97511692376801089092939793109, −6.07887769071597852891685589833, −5.58745441732081187675374765932, −4.45270117982561523167969303298, −3.14863762182812626778293578974, −1.64836909204780835784659888288, 0.04668180101588838862112320675, 1.36368402738150094671511568627, 3.34055905477455114835888612817, 4.24186606348388951505734625968, 5.26262404835290895004513423193, 6.52636517926988765590329188232, 6.81664260049868124871848008162, 7.86804252619135832064978638615, 9.215064380979145247710773270005, 10.04340955618697649378052719973

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