Properties

Label 2-684-19.18-c6-0-29
Degree $2$
Conductor $684$
Sign $0.398 + 0.916i$
Analytic cond. $157.356$
Root an. cond. $12.5442$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 102.·5-s + 512.·7-s − 2.41e3·11-s + 3.77e3i·13-s − 1.56e3·17-s + (−2.73e3 − 6.28e3i)19-s + 626.·23-s − 5.12e3·25-s + 1.57e4i·29-s + 2.61e4i·31-s − 5.25e4·35-s − 4.94e4i·37-s − 1.19e5i·41-s − 1.33e4·43-s + 1.86e5·47-s + ⋯
L(s)  = 1  − 0.819·5-s + 1.49·7-s − 1.81·11-s + 1.71i·13-s − 0.318·17-s + (−0.398 − 0.916i)19-s + 0.0514·23-s − 0.327·25-s + 0.643i·29-s + 0.879i·31-s − 1.22·35-s − 0.977i·37-s − 1.73i·41-s − 0.167·43-s + 1.79·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.398 + 0.916i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.398 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.398 + 0.916i$
Analytic conductor: \(157.356\)
Root analytic conductor: \(12.5442\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :3),\ 0.398 + 0.916i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.016935816\)
\(L(\frac12)\) \(\approx\) \(1.016935816\)
\(L(4)\) 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 \( 1 + (2.73e3 + 6.28e3i)T \)
good5 \( 1 + 102.T + 1.56e4T^{2} \)
7 \( 1 - 512.T + 1.17e5T^{2} \)
11 \( 1 + 2.41e3T + 1.77e6T^{2} \)
13 \( 1 - 3.77e3iT - 4.82e6T^{2} \)
17 \( 1 + 1.56e3T + 2.41e7T^{2} \)
23 \( 1 - 626.T + 1.48e8T^{2} \)
29 \( 1 - 1.57e4iT - 5.94e8T^{2} \)
31 \( 1 - 2.61e4iT - 8.87e8T^{2} \)
37 \( 1 + 4.94e4iT - 2.56e9T^{2} \)
41 \( 1 + 1.19e5iT - 4.75e9T^{2} \)
43 \( 1 + 1.33e4T + 6.32e9T^{2} \)
47 \( 1 - 1.86e5T + 1.07e10T^{2} \)
53 \( 1 - 2.04e5iT - 2.21e10T^{2} \)
59 \( 1 + 1.82e4iT - 4.21e10T^{2} \)
61 \( 1 + 3.53e5T + 5.15e10T^{2} \)
67 \( 1 - 5.19e5iT - 9.04e10T^{2} \)
71 \( 1 + 2.68e5iT - 1.28e11T^{2} \)
73 \( 1 - 5.85e4T + 1.51e11T^{2} \)
79 \( 1 + 4.73e5iT - 2.43e11T^{2} \)
83 \( 1 + 1.27e5T + 3.26e11T^{2} \)
89 \( 1 - 6.07e5iT - 4.96e11T^{2} \)
97 \( 1 - 2.82e5iT - 8.32e11T^{2} \)
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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

−9.087629195741339052954415294173, −8.529373727435374724291624764527, −7.55580671405639815669177465247, −7.11598763741344012882889290471, −5.58383594429893646218262117412, −4.71465716648521887217829650152, −4.11296033005248403178004673604, −2.58200177040045551487701198538, −1.72236689999462615822121354678, −0.26239215391494756346910061296, 0.72152059225821554595429207517, 2.09430235568085717821704356003, 3.11092782245147642375073121061, 4.35626429566158548068918532793, 5.13944053009857810579007527028, 5.92299122104712079644198648645, 7.59768015110059507578143165745, 7.994776792549016330022701615777, 8.303738757621816393528161233593, 9.938843267233061682708720566135

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