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Properties

Label 2-684-171.103-c2-0-12
Degree $2$
Conductor $684$
Sign $0.738 - 0.673i$
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  + (2.41 − 1.78i)3-s + (−0.193 − 0.335i)5-s + (5.33 + 9.24i)7-s + (2.66 − 8.59i)9-s + (1.58 + 2.75i)11-s + 16.0i·13-s + (−1.06 − 0.465i)15-s + (−14.5 + 25.2i)17-s + (−5.25 + 18.2i)19-s + (29.3 + 12.8i)21-s + 16.4·23-s + (12.4 − 21.5i)25-s + (−8.88 − 25.4i)27-s + (−33.8 − 19.5i)29-s + (31.7 + 18.3i)31-s + ⋯
L(s)  = 1  + (0.804 − 0.593i)3-s + (−0.0387 − 0.0671i)5-s + (0.762 + 1.32i)7-s + (0.295 − 0.955i)9-s + (0.144 + 0.250i)11-s + 1.23i·13-s + (−0.0710 − 0.0310i)15-s + (−0.855 + 1.48i)17-s + (−0.276 + 0.960i)19-s + (1.39 + 0.610i)21-s + 0.715·23-s + (0.496 − 0.860i)25-s + (−0.329 − 0.944i)27-s + (−1.16 − 0.674i)29-s + (1.02 + 0.591i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 - 0.673i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.738 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.738 - 0.673i$
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: \((2,\ 684,\ (\ :1),\ 0.738 - 0.673i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.471661546\)
\(L(\frac12)\) \(\approx\) \(2.471661546\)
\(L(2)\) 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 \( 1 + (-2.41 + 1.78i)T \)
19 \( 1 + (5.25 - 18.2i)T \)
good5 \( 1 + (0.193 + 0.335i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-5.33 - 9.24i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-1.58 - 2.75i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 16.0iT - 169T^{2} \)
17 \( 1 + (14.5 - 25.2i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 - 16.4T + 529T^{2} \)
29 \( 1 + (33.8 + 19.5i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-31.7 - 18.3i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 30.9iT - 1.36e3T^{2} \)
41 \( 1 + (30.2 - 17.4i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + 14.1T + 1.84e3T^{2} \)
47 \( 1 + (-13.6 + 23.7i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-40.2 + 23.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-59.9 + 34.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-27.4 + 47.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 - 8.73iT - 4.48e3T^{2} \)
71 \( 1 + (18.3 + 10.5i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-21.9 + 37.9i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 70.3iT - 6.24e3T^{2} \)
83 \( 1 + (-42.8 - 74.2i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-96.5 + 55.7i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 157. iT - 9.40e3T^{2} \)
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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

−10.26134893742049740470980472349, −9.208240759665250122697456968845, −8.526028932210812323531780456052, −8.121649597977391354616400302931, −6.77080113319347618042070206172, −6.15211919983992156599624682990, −4.80227590061421945163193237818, −3.76073244649470914467223243467, −2.27548876647676199743843800483, −1.68163641151211782198589136243, 0.825465466065677922734590685024, 2.53312066541373037110541517088, 3.57024660950037901714777773793, 4.59272285391242327458356698720, 5.29111370701282768216971134686, 7.11280989233261426507001702522, 7.44656963725493058676416286347, 8.555785970302605229555128732287, 9.233183589023562787547198971424, 10.25813907020201093405089786320

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