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Properties

Label 2-684-19.18-c2-0-13
Degree $2$
Conductor $684$
Sign $0.830 + 0.556i$
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  + 7.39·5-s + 3.28·7-s + 0.714·11-s − 25.4i·13-s + 26.0·17-s + (−15.7 − 10.5i)19-s − 8.46·23-s + 29.6·25-s + 23.1i·29-s − 18.8i·31-s + 24.2·35-s + 48.5i·37-s − 35.3i·41-s − 24.5·43-s + 46.0·47-s + ⋯
L(s)  = 1  + 1.47·5-s + 0.469·7-s + 0.0649·11-s − 1.95i·13-s + 1.53·17-s + (−0.830 − 0.556i)19-s − 0.367·23-s + 1.18·25-s + 0.797i·29-s − 0.608i·31-s + 0.693·35-s + 1.31i·37-s − 0.862i·41-s − 0.569·43-s + 0.980·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.830 + 0.556i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ 0.830 + 0.556i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.566689171\)
\(L(\frac12)\) \(\approx\) \(2.566689171\)
\(L(2)\) 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 \( 1 \)
19 \( 1 + (15.7 + 10.5i)T \)
good5 \( 1 - 7.39T + 25T^{2} \)
7 \( 1 - 3.28T + 49T^{2} \)
11 \( 1 - 0.714T + 121T^{2} \)
13 \( 1 + 25.4iT - 169T^{2} \)
17 \( 1 - 26.0T + 289T^{2} \)
23 \( 1 + 8.46T + 529T^{2} \)
29 \( 1 - 23.1iT - 841T^{2} \)
31 \( 1 + 18.8iT - 961T^{2} \)
37 \( 1 - 48.5iT - 1.36e3T^{2} \)
41 \( 1 + 35.3iT - 1.68e3T^{2} \)
43 \( 1 + 24.5T + 1.84e3T^{2} \)
47 \( 1 - 46.0T + 2.20e3T^{2} \)
53 \( 1 + 51.2iT - 2.80e3T^{2} \)
59 \( 1 - 101. iT - 3.48e3T^{2} \)
61 \( 1 - 38.5T + 3.72e3T^{2} \)
67 \( 1 + 44.2iT - 4.48e3T^{2} \)
71 \( 1 + 74.3iT - 5.04e3T^{2} \)
73 \( 1 - 112.T + 5.32e3T^{2} \)
79 \( 1 - 125. iT - 6.24e3T^{2} \)
83 \( 1 - 95.5T + 6.88e3T^{2} \)
89 \( 1 + 28.0iT - 7.92e3T^{2} \)
97 \( 1 + 55.7iT - 9.40e3T^{2} \)
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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

−10.27119060813843893371559246135, −9.502712555475599814690568474267, −8.439625753130001790953569241305, −7.70197456785041739556740675123, −6.45781206207610688829915934091, −5.58387741284985875985248673019, −5.06080286541129116865412065118, −3.38926347277523642601927325913, −2.29429480528579978870281346760, −1.01506118579906103402320325921, 1.49925819886774601209141451855, 2.25085842209378547523154471306, 3.87252203591956780865057996283, 4.98715213326277387043380671541, 5.95716937235358077907171119086, 6.60504673494830915214673886671, 7.76543284486206314371764483323, 8.816237751733403505523470927646, 9.589637539777819251659999132811, 10.15377921663920683185846742213

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