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Properties

Label 2-684-171.56-c1-0-18
Degree $2$
Conductor $684$
Sign $0.752 + 0.658i$
Analytic cond. $5.46176$
Root an. cond. $2.33704$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.72 − 0.192i)3-s + (2.63 − 1.52i)5-s + (0.965 − 1.67i)7-s + (2.92 − 0.664i)9-s + (−1.62 − 0.937i)11-s + (−0.969 + 0.559i)13-s + (4.24 − 3.13i)15-s + 0.294i·17-s + (−4.18 + 1.20i)19-s + (1.33 − 3.06i)21-s + (−1.45 + 0.838i)23-s + (2.14 − 3.71i)25-s + (4.90 − 1.70i)27-s + (−2.02 + 3.50i)29-s + (−6.92 + 3.99i)31-s + ⋯
L(s)  = 1  + (0.993 − 0.111i)3-s + (1.18 − 0.681i)5-s + (0.364 − 0.631i)7-s + (0.975 − 0.221i)9-s + (−0.489 − 0.282i)11-s + (−0.269 + 0.155i)13-s + (1.09 − 0.808i)15-s + 0.0714i·17-s + (−0.961 + 0.276i)19-s + (0.292 − 0.668i)21-s + (−0.302 + 0.174i)23-s + (0.428 − 0.742i)25-s + (0.944 − 0.328i)27-s + (−0.376 + 0.651i)29-s + (−1.24 + 0.718i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.752 + 0.658i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ 0.752 + 0.658i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.36464 - 0.888557i\)
\(L(\frac12)\) \(\approx\) \(2.36464 - 0.888557i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.192i)T \)
19 \( 1 + (4.18 - 1.20i)T \)
good5 \( 1 + (-2.63 + 1.52i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.965 + 1.67i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.62 + 0.937i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.969 - 0.559i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.294iT - 17T^{2} \)
23 \( 1 + (1.45 - 0.838i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.02 - 3.50i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.92 - 3.99i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.16iT - 37T^{2} \)
41 \( 1 + (-3.59 - 6.21i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.53 + 6.12i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.28 - 1.89i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.633T + 53T^{2} \)
59 \( 1 + (-2.99 - 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.58 + 2.74i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.82 - 1.05i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 - 5.01T + 73T^{2} \)
79 \( 1 + (11.0 + 6.35i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.47 - 1.43i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 9.69T + 89T^{2} \)
97 \( 1 + (7.04 + 4.06i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.25066700677348459480783695883, −9.389244409971980425966595078052, −8.765358355745506006527880257215, −7.891209983172533037657080053594, −7.00336119043570358651583201130, −5.84683600571133228127196612834, −4.83974021839476596143883437663, −3.78587055058610030315201737705, −2.37815161050343582732347013167, −1.41544051556181599673617911828, 2.14085720904496137628004784840, 2.44801845562902772914245819808, 3.92118923954161076794955367451, 5.18648380185948943865223661560, 6.11633153996349052870797568124, 7.16317136820081145303822881254, 8.034296032878307837968056822852, 8.994627346322245234521305691666, 9.657205576477233289125632605953, 10.38836636569466034799124560291

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