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Properties

Label 2-693-11.4-c1-0-1
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: $\chi_{693} (631, \cdot )$
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

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

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