Properties

Label 2-684-171.11-c2-0-3
Degree $2$
Conductor $684$
Sign $-0.311 + 0.950i$
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  + (−1.55 + 2.56i)3-s + 5.60i·5-s + (−5.05 + 8.75i)7-s + (−4.18 − 7.96i)9-s + (−4.84 − 2.79i)11-s + (−9.41 + 16.3i)13-s + (−14.3 − 8.69i)15-s + (1.70 + 0.983i)17-s + (−12.6 − 14.1i)19-s + (−14.6 − 26.5i)21-s + (14.6 + 8.46i)23-s − 6.39·25-s + (26.9 + 1.60i)27-s + 12.2i·29-s + (3.88 + 6.73i)31-s + ⋯
L(s)  = 1  + (−0.517 + 0.855i)3-s + 1.12i·5-s + (−0.722 + 1.25i)7-s + (−0.465 − 0.885i)9-s + (−0.440 − 0.254i)11-s + (−0.724 + 1.25i)13-s + (−0.959 − 0.579i)15-s + (0.100 + 0.0578i)17-s + (−0.667 − 0.744i)19-s + (−0.697 − 1.26i)21-s + (0.637 + 0.367i)23-s − 0.255·25-s + (0.998 + 0.0593i)27-s + 0.423i·29-s + (0.125 + 0.217i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.311 + 0.950i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.311 + 0.950i$
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: \((2,\ 684,\ (\ :1),\ -0.311 + 0.950i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4980763811\)
\(L(\frac12)\) \(\approx\) \(0.4980763811\)
\(L(2)\) 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 \( 1 + (1.55 - 2.56i)T \)
19 \( 1 + (12.6 + 14.1i)T \)
good5 \( 1 - 5.60iT - 25T^{2} \)
7 \( 1 + (5.05 - 8.75i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (4.84 + 2.79i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (9.41 - 16.3i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-1.70 - 0.983i)T + (144.5 + 250. i)T^{2} \)
23 \( 1 + (-14.6 - 8.46i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 12.2iT - 841T^{2} \)
31 \( 1 + (-3.88 - 6.73i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 47.4T + 1.36e3T^{2} \)
41 \( 1 - 36.8iT - 1.68e3T^{2} \)
43 \( 1 + (20.2 + 35.1i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + 1.53iT - 2.20e3T^{2} \)
53 \( 1 + (-15.7 + 9.06i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + 25.1iT - 3.48e3T^{2} \)
61 \( 1 - 63.5T + 3.72e3T^{2} \)
67 \( 1 + (-12.0 + 20.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (106. + 61.3i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (10.0 - 17.3i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-5.16 - 8.94i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (39.0 + 22.5i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (141. - 81.7i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (90.0 + 155. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.93922373541686272809797920398, −9.975427964513574189597963217122, −9.347310138597621986422279109392, −8.587052424828148798787539754764, −7.02433309231635150295227984416, −6.43969800318528638412182347810, −5.53509151091467494747062815390, −4.52623897613307637578470805032, −3.20770489442746380796753077891, −2.47794487601585765653881395939, 0.21678432147718878988383308735, 1.08671053154208834213633560193, 2.70391840376181631986955154912, 4.22918062434636720100437826166, 5.16647000468869747655366402152, 6.06573226128032936099556863390, 7.14678952049591678197095074467, 7.78886933866610973372849352425, 8.598451263614908189072997319514, 9.940410237646269575730871117293

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