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Properties

Label 2-693-11.3-c1-0-25
Degree $2$
Conductor $693$
Sign $-0.610 + 0.792i$
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  + (0.614 − 0.446i)2-s + (−0.439 + 1.35i)4-s + (−2.31 − 1.68i)5-s + (0.309 − 0.951i)7-s + (0.803 + 2.47i)8-s − 2.17·10-s + (−3.15 + 1.01i)11-s + (5.63 − 4.09i)13-s + (−0.234 − 0.722i)14-s + (−0.706 − 0.512i)16-s + (−5.38 − 3.91i)17-s + (−1.77 − 5.46i)19-s + (3.29 − 2.39i)20-s + (−1.48 + 2.03i)22-s − 0.724·23-s + ⋯
L(s)  = 1  + (0.434 − 0.315i)2-s + (−0.219 + 0.676i)4-s + (−1.03 − 0.753i)5-s + (0.116 − 0.359i)7-s + (0.284 + 0.874i)8-s − 0.688·10-s + (−0.952 + 0.304i)11-s + (1.56 − 1.13i)13-s + (−0.0627 − 0.193i)14-s + (−0.176 − 0.128i)16-s + (−1.30 − 0.949i)17-s + (−0.407 − 1.25i)19-s + (0.737 − 0.536i)20-s + (−0.317 + 0.432i)22-s − 0.151·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.610 + 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.610 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.610 + 0.792i$
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: \((2,\ 693,\ (\ :1/2),\ -0.610 + 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.393361 - 0.799783i\)
\(L(\frac12)\) \(\approx\) \(0.393361 - 0.799783i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.15 - 1.01i)T \)
good2 \( 1 + (-0.614 + 0.446i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (2.31 + 1.68i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-5.63 + 4.09i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (5.38 + 3.91i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.77 + 5.46i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 0.724T + 23T^{2} \)
29 \( 1 + (-2.33 + 7.19i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.28 - 1.65i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.532 - 1.63i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.346 - 1.06i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 11.7T + 43T^{2} \)
47 \( 1 + (-1.60 - 4.93i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-5.06 + 3.67i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.23 + 3.78i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-4.83 - 3.50i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 + (-0.805 - 0.584i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.09 + 3.36i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-0.541 + 0.393i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (10.8 + 7.89i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 16.6T + 89T^{2} \)
97 \( 1 + (8.18 - 5.94i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.43624447628405764579181128403, −9.010029307308559328784437069273, −8.291773861812770179546743801671, −7.84926839401783893359043726699, −6.73554896244312842295243668663, −5.17704795657884364150823973984, −4.53064737342107884294678554245, −3.66846771158932952773271959919, −2.56271143413745415391404330660, −0.40488001415352502758134798206, 1.81555038388660092052977794516, 3.59134834955710775689511507534, 4.18500206308765232143434344460, 5.45494455408204456156757762865, 6.36474068665762143183122669798, 6.98173316620349757403862663698, 8.292645271122144713494189710829, 8.773007049137622737697070463323, 10.15200369540645776166117922200, 10.90428542383880471719531447679

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