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Properties

Label 2-684-57.11-c2-0-10
Degree $2$
Conductor $684$
Sign $-0.877 + 0.478i$
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  + (−3.59 − 2.07i)5-s + 4.23·7-s + 14.0i·11-s + (−6.49 − 11.2i)13-s + (−8.58 − 4.95i)17-s + (6.13 − 17.9i)19-s + (−1.94 + 1.12i)23-s + (−3.87 − 6.71i)25-s + (−3.25 + 1.87i)29-s + 15.9·31-s + (−15.2 − 8.78i)35-s − 21.7·37-s + (−53.3 − 30.8i)41-s + (−18.1 + 31.4i)43-s + (−48.8 + 28.2i)47-s + ⋯
L(s)  = 1  + (−0.719 − 0.415i)5-s + 0.604·7-s + 1.27i·11-s + (−0.499 − 0.865i)13-s + (−0.504 − 0.291i)17-s + (0.322 − 0.946i)19-s + (−0.0843 + 0.0487i)23-s + (−0.155 − 0.268i)25-s + (−0.112 + 0.0647i)29-s + 0.514·31-s + (−0.434 − 0.250i)35-s − 0.586·37-s + (−1.30 − 0.751i)41-s + (−0.422 + 0.731i)43-s + (−1.03 + 0.600i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.877 + 0.478i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.877 + 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.877 + 0.478i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ -0.877 + 0.478i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5362316601\)
\(L(\frac12)\) \(\approx\) \(0.5362316601\)
\(L(2)\) 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 \( 1 + (-6.13 + 17.9i)T \)
good5 \( 1 + (3.59 + 2.07i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 - 4.23T + 49T^{2} \)
11 \( 1 - 14.0iT - 121T^{2} \)
13 \( 1 + (6.49 + 11.2i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (8.58 + 4.95i)T + (144.5 + 250. i)T^{2} \)
23 \( 1 + (1.94 - 1.12i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (3.25 - 1.87i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 15.9T + 961T^{2} \)
37 \( 1 + 21.7T + 1.36e3T^{2} \)
41 \( 1 + (53.3 + 30.8i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (18.1 - 31.4i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (48.8 - 28.2i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (21.9 - 12.6i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (41.1 + 23.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (22.8 + 39.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (37.6 + 65.1i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (97.0 + 56.0i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-62.8 + 108. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-4.87 + 8.44i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 40.9iT - 6.88e3T^{2} \)
89 \( 1 + (-95.2 + 55.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (5.36 - 9.29i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−9.902710674643740960986780244059, −9.040186835503536013892550134506, −8.014916290477401980294659964708, −7.49653596914322600664542017117, −6.47231717903676387816334629680, −4.84391981654008338943816746648, −4.74939851321779852838836188553, −3.21755265888111536841887566703, −1.85045128970373204702853696174, −0.18739884485117735857302497770, 1.63786807186899807855583237230, 3.15375409180858944114302338839, 4.05931379984884000151009442470, 5.16496643506043279386753736656, 6.26819029408520691867065509302, 7.18211442129040373584115814272, 8.126687264739491024362143759449, 8.689419698451772038004609499256, 9.868361459268113012765952475923, 10.75475732836087785761328020272

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