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Properties

Label 2-684-171.164-c1-0-13
Degree $2$
Conductor $684$
Sign $-0.203 + 0.979i$
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  + (−0.237 − 1.71i)3-s + (1.87 + 1.08i)5-s + (−0.579 + 1.00i)7-s + (−2.88 + 0.814i)9-s + (−4.75 − 2.74i)11-s − 3.68i·13-s + (1.41 − 3.47i)15-s + (6.21 − 3.58i)17-s + (2.91 − 3.24i)19-s + (1.86 + 0.756i)21-s − 7.99i·23-s + (−0.160 − 0.277i)25-s + (2.08 + 4.76i)27-s + (1.63 + 2.83i)29-s + (−3.80 + 2.19i)31-s + ⋯
L(s)  = 1  + (−0.137 − 0.990i)3-s + (0.837 + 0.483i)5-s + (−0.219 + 0.379i)7-s + (−0.962 + 0.271i)9-s + (−1.43 − 0.827i)11-s − 1.02i·13-s + (0.364 − 0.896i)15-s + (1.50 − 0.870i)17-s + (0.667 − 0.744i)19-s + (0.405 + 0.164i)21-s − 1.66i·23-s + (−0.0320 − 0.0555i)25-s + (0.400 + 0.916i)27-s + (0.304 + 0.526i)29-s + (−0.683 + 0.394i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.203 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.203 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.203 + 0.979i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (677, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ -0.203 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.842068 - 1.03500i\)
\(L(\frac12)\) \(\approx\) \(0.842068 - 1.03500i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.237 + 1.71i)T \)
19 \( 1 + (-2.91 + 3.24i)T \)
good5 \( 1 + (-1.87 - 1.08i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.579 - 1.00i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.75 + 2.74i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.68iT - 13T^{2} \)
17 \( 1 + (-6.21 + 3.58i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + 7.99iT - 23T^{2} \)
29 \( 1 + (-1.63 - 2.83i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.80 - 2.19i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.39iT - 37T^{2} \)
41 \( 1 + (-1.98 + 3.43i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 6.78T + 43T^{2} \)
47 \( 1 + (6.01 - 3.47i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.23 + 3.86i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.20 + 2.08i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.52 - 2.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 - 10.2iT - 67T^{2} \)
71 \( 1 + (7.79 + 13.4i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.86 - 4.96i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 6.33iT - 79T^{2} \)
83 \( 1 + (0.385 + 0.222i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.64 - 9.78i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.9iT - 97T^{2} \)
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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

−10.46305280866126256367402183376, −9.379026529322678479634678380928, −8.301532960844760261370375132161, −7.62273849579265211285914059790, −6.67648012278646270220558454177, −5.58525259225815610800889810832, −5.36968232088046899087411172467, −2.92477630354062082943602695700, −2.63364949998296554527330253932, −0.71999466438231547018831760065, 1.73221594459796518451682126425, 3.26586105271109163537947922510, 4.31581002556843588051402928493, 5.44670730936316053063173653051, 5.78387567608750401749915976694, 7.34318611331899686720091737732, 8.168399758691154211742984905738, 9.454455172915143318211206739691, 9.807347404362823654124463290645, 10.39029429939157245898405951196

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