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Properties

Label 2-684-12.11-c1-0-24
Degree $2$
Conductor $684$
Sign $0.992 + 0.123i$
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  + (1.37 − 0.344i)2-s + (1.76 − 0.944i)4-s + 2.95i·5-s − 2.52i·7-s + (2.09 − 1.90i)8-s + (1.01 + 4.04i)10-s + 0.275·11-s + 3.94·13-s + (−0.869 − 3.46i)14-s + (2.21 − 3.33i)16-s + 2.91i·17-s − i·19-s + (2.78 + 5.20i)20-s + (0.378 − 0.0949i)22-s + 5.02·23-s + ⋯
L(s)  = 1  + (0.969 − 0.243i)2-s + (0.881 − 0.472i)4-s + 1.32i·5-s − 0.954i·7-s + (0.739 − 0.672i)8-s + (0.321 + 1.28i)10-s + 0.0831·11-s + 1.09·13-s + (−0.232 − 0.925i)14-s + (0.553 − 0.832i)16-s + 0.705i·17-s − 0.229i·19-s + (0.623 + 1.16i)20-s + (0.0806 − 0.0202i)22-s + 1.04·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.992 + 0.123i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ 0.992 + 0.123i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.88988 - 0.178883i\)
\(L(\frac12)\) \(\approx\) \(2.88988 - 0.178883i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 + 0.344i)T \)
3 \( 1 \)
19 \( 1 + iT \)
good5 \( 1 - 2.95iT - 5T^{2} \)
7 \( 1 + 2.52iT - 7T^{2} \)
11 \( 1 - 0.275T + 11T^{2} \)
13 \( 1 - 3.94T + 13T^{2} \)
17 \( 1 - 2.91iT - 17T^{2} \)
23 \( 1 - 5.02T + 23T^{2} \)
29 \( 1 - 0.0951iT - 29T^{2} \)
31 \( 1 - 6.84iT - 31T^{2} \)
37 \( 1 + 11.9T + 37T^{2} \)
41 \( 1 - 4.10iT - 41T^{2} \)
43 \( 1 + 8.64iT - 43T^{2} \)
47 \( 1 + 5.81T + 47T^{2} \)
53 \( 1 + 0.551iT - 53T^{2} \)
59 \( 1 + 5.80T + 59T^{2} \)
61 \( 1 + 13.7T + 61T^{2} \)
67 \( 1 + 6.79iT - 67T^{2} \)
71 \( 1 - 8.34T + 71T^{2} \)
73 \( 1 + 3.65T + 73T^{2} \)
79 \( 1 + 9.13iT - 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 + 6.30iT - 89T^{2} \)
97 \( 1 - 0.763T + 97T^{2} \)
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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

−10.74821679327244452134271820002, −10.14442260149979852217475186880, −8.712221345886193745570588004910, −7.38191048795186847209858647935, −6.81695483217862852003306148776, −6.10562844016343116042050653220, −4.85588041464503893780418362018, −3.65214667678954831093451745658, −3.14597322208854791555120334976, −1.56653251702606900795405357843, 1.52219117956209601820314985608, 2.95948241593785501681251154752, 4.17833296439237396779152455461, 5.13732499032697706046255705526, 5.71538674858311628197202988987, 6.73069387821868099721507784205, 7.948346766210470992815764196178, 8.704345755614482617912077904321, 9.367172082340454290400867384427, 10.79644951244312153232977921151

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