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Properties

Label 2-684-19.13-c0-0-0
Degree $2$
Conductor $684$
Sign $0.500 - 0.865i$
Analytic cond. $0.341360$
Root an. cond. $0.584260$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.939 + 1.62i)7-s + (0.673 + 1.85i)13-s + (0.5 − 0.866i)19-s + (0.939 − 0.342i)25-s + (−1.11 − 0.642i)31-s − 1.28i·37-s + (0.266 + 1.50i)43-s + (−1.26 − 2.19i)49-s + (−0.0603 + 0.342i)61-s + (0.439 − 0.524i)67-s + (−0.326 − 0.118i)73-s + (−0.233 + 0.642i)79-s + (−3.64 − 0.642i)91-s + (−1.11 − 1.32i)97-s + (1.70 − 0.984i)103-s + ⋯
L(s)  = 1  + (−0.939 + 1.62i)7-s + (0.673 + 1.85i)13-s + (0.5 − 0.866i)19-s + (0.939 − 0.342i)25-s + (−1.11 − 0.642i)31-s − 1.28i·37-s + (0.266 + 1.50i)43-s + (−1.26 − 2.19i)49-s + (−0.0603 + 0.342i)61-s + (0.439 − 0.524i)67-s + (−0.326 − 0.118i)73-s + (−0.233 + 0.642i)79-s + (−3.64 − 0.642i)91-s + (−1.11 − 1.32i)97-s + (1.70 − 0.984i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.500 - 0.865i$
Analytic conductor: \(0.341360\)
Root analytic conductor: \(0.584260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :0),\ 0.500 - 0.865i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8562109469\)
\(L(\frac12)\) \(\approx\) \(0.8562109469\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-0.939 + 0.342i)T^{2} \)
7 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.673 - 1.85i)T + (-0.766 + 0.642i)T^{2} \)
17 \( 1 + (0.173 - 0.984i)T^{2} \)
23 \( 1 + (-0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 - 0.984i)T^{2} \)
31 \( 1 + (1.11 + 0.642i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + 1.28iT - T^{2} \)
41 \( 1 + (-0.766 - 0.642i)T^{2} \)
43 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (0.173 + 0.984i)T^{2} \)
53 \( 1 + (0.939 + 0.342i)T^{2} \)
59 \( 1 + (-0.173 + 0.984i)T^{2} \)
61 \( 1 + (0.0603 - 0.342i)T + (-0.939 - 0.342i)T^{2} \)
67 \( 1 + (-0.439 + 0.524i)T + (-0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
79 \( 1 + (0.233 - 0.642i)T + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.766 + 0.642i)T^{2} \)
97 \( 1 + (1.11 + 1.32i)T + (-0.173 + 0.984i)T^{2} \)
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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

−11.06352922953166154869721830929, −9.609197801806292359133098823561, −9.172762310488682760614768565863, −8.572535651115436402659249595746, −7.13664619692515108186097279211, −6.36038214156932822595908584551, −5.58901612498828482792561275372, −4.38516822615085000261171675606, −3.11518264521038519215997492338, −2.05313967395021097894259951382, 1.02671385901535761993912402830, 3.20681298209757081057277061821, 3.72134879886700528736049903073, 5.12884487058737318419991300634, 6.14605667666158444645899674368, 7.12547063140059893595785253814, 7.78880347985238086318905752109, 8.807068164156596689130966175959, 10.04150274772410226642764724140, 10.37364632754520356738571892985

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