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Properties

Label 2-693-11.3-c1-0-27
Degree $2$
Conductor $693$
Sign $-0.266 + 0.963i$
Analytic cond. $5.53363$
Root an. cond. $2.35236$
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  + (2.11 − 1.53i)2-s + (1.5 − 4.61i)4-s + (1.30 + 0.951i)5-s + (0.309 − 0.951i)7-s + (−2.30 − 7.10i)8-s + 4.23·10-s + (2.54 − 2.12i)11-s + (−4.42 + 3.21i)13-s + (−0.809 − 2.48i)14-s + (−7.97 − 5.79i)16-s + (−2.42 − 1.76i)17-s + (1.23 + 3.80i)19-s + (6.35 − 4.61i)20-s + (2.11 − 8.42i)22-s + 7.23·23-s + ⋯
L(s)  = 1  + (1.49 − 1.08i)2-s + (0.750 − 2.30i)4-s + (0.585 + 0.425i)5-s + (0.116 − 0.359i)7-s + (−0.816 − 2.51i)8-s + 1.33·10-s + (0.767 − 0.641i)11-s + (−1.22 + 0.892i)13-s + (−0.216 − 0.665i)14-s + (−1.99 − 1.44i)16-s + (−0.588 − 0.427i)17-s + (0.283 + 0.872i)19-s + (1.42 − 1.03i)20-s + (0.451 − 1.79i)22-s + 1.50·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.266 + 0.963i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ -0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.22258 - 2.92181i\)
\(L(\frac12)\) \(\approx\) \(2.22258 - 2.92181i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (-2.54 + 2.12i)T \)
good2 \( 1 + (-2.11 + 1.53i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-1.30 - 0.951i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (4.42 - 3.21i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.42 + 1.76i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.23 - 3.80i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 7.23T + 23T^{2} \)
29 \( 1 + (0.927 - 2.85i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.190 - 0.138i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.61 - 8.05i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.954 - 2.93i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 9T + 43T^{2} \)
47 \( 1 + (3.11 + 9.59i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.30 + 2.40i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.89 - 12.0i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-5.42 - 3.94i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 3.70T + 67T^{2} \)
71 \( 1 + (-9.89 - 7.19i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.19 - 3.66i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.54 - 4.02i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.38 - 1.73i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 2.67T + 89T^{2} \)
97 \( 1 + (-13.2 + 9.64i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.32113906382326552568638159163, −9.879911311459249896199147254257, −8.811132243034520649045510365134, −7.01510017822673398291664852410, −6.48932355412077503436307665251, −5.35041822008848363493571201286, −4.59028950169664960916953918769, −3.54322539896591095158460373465, −2.57593912884776868276648180604, −1.45405364077196737883280359672, 2.24734104049669135045988994480, 3.43123611625457520332289214893, 4.77863916114573612371415108382, 5.11089861938665868970943022075, 6.13027513094980032805844073879, 7.00881250528793315637423966242, 7.68168320545103570469873250105, 8.860694851490151230900202399845, 9.591973204094583376077052283146, 11.02916629560942244689768205942

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