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Properties

Label 2-684-228.11-c1-0-7
Degree $2$
Conductor $684$
Sign $0.840 - 0.541i$
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  + (−1.16 + 0.798i)2-s + (0.724 − 1.86i)4-s + (−0.436 − 0.252i)5-s − 1.18i·7-s + (0.642 + 2.75i)8-s + (0.711 − 0.0544i)10-s − 5.00·11-s + (2.43 + 4.22i)13-s + (0.949 + 1.38i)14-s + (−2.95 − 2.70i)16-s + (3.84 + 2.22i)17-s + (4.16 − 1.28i)19-s + (−0.786 + 0.631i)20-s + (5.84 − 3.99i)22-s + (2.90 + 5.04i)23-s + ⋯
L(s)  = 1  + (−0.825 + 0.564i)2-s + (0.362 − 0.932i)4-s + (−0.195 − 0.112i)5-s − 0.449i·7-s + (0.227 + 0.973i)8-s + (0.224 − 0.0172i)10-s − 1.50·11-s + (0.676 + 1.17i)13-s + (0.253 + 0.370i)14-s + (−0.737 − 0.675i)16-s + (0.933 + 0.539i)17-s + (0.955 − 0.293i)19-s + (−0.175 + 0.141i)20-s + (1.24 − 0.851i)22-s + (0.606 + 1.05i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.840 - 0.541i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ 0.840 - 0.541i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.908718 + 0.267226i\)
\(L(\frac12)\) \(\approx\) \(0.908718 + 0.267226i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.16 - 0.798i)T \)
3 \( 1 \)
19 \( 1 + (-4.16 + 1.28i)T \)
good5 \( 1 + (0.436 + 0.252i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 1.18iT - 7T^{2} \)
11 \( 1 + 5.00T + 11T^{2} \)
13 \( 1 + (-2.43 - 4.22i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.84 - 2.22i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.90 - 5.04i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-8.28 + 4.78i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.47iT - 31T^{2} \)
37 \( 1 - 4.92T + 37T^{2} \)
41 \( 1 + (-2.73 - 1.57i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-7.91 - 4.56i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.62 + 9.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.24 + 1.87i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.302 - 0.523i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.31 - 9.20i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.58 - 3.22i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.407 - 0.706i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.13 - 10.6i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.41 + 1.97i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.97T + 83T^{2} \)
89 \( 1 + (2.55 - 1.47i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.01 + 1.75i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.26920029767132603276026739248, −9.780757540662144225328847155593, −8.701380438762559110819559252797, −7.88506630453306806336763917682, −7.32876749903417153995972967936, −6.19732209677283298540830940177, −5.35776593475416362134400052550, −4.20381197035951643580557999221, −2.59862272965867261751865920459, −1.01029650864122925781674728577, 0.911859284706373815885228521092, 2.72622412721696572678584221962, 3.26554267856784101111608645832, 4.94087163085792498896591227916, 5.91133091487580885780501609790, 7.32428149931015486931494750774, 7.901071776617798303686186849409, 8.656988442441997432689647689048, 9.644241600056552404641258939088, 10.52868590206053048489673778756

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