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Properties

Label 2-693-11.3-c1-0-15
Degree $2$
Conductor $693$
Sign $0.838 - 0.544i$
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  + (0.901 − 0.655i)2-s + (−0.234 + 0.720i)4-s + (2.79 + 2.03i)5-s + (0.309 − 0.951i)7-s + (0.949 + 2.92i)8-s + 3.85·10-s + (−3.31 − 0.0938i)11-s + (1.66 − 1.21i)13-s + (−0.344 − 1.05i)14-s + (1.54 + 1.12i)16-s + (1.56 + 1.13i)17-s + (0.501 + 1.54i)19-s + (−2.11 + 1.53i)20-s + (−3.05 + 2.08i)22-s + 0.807·23-s + ⋯
L(s)  = 1  + (0.637 − 0.463i)2-s + (−0.117 + 0.360i)4-s + (1.25 + 0.908i)5-s + (0.116 − 0.359i)7-s + (0.335 + 1.03i)8-s + 1.21·10-s + (−0.999 − 0.0283i)11-s + (0.462 − 0.335i)13-s + (−0.0920 − 0.283i)14-s + (0.386 + 0.280i)16-s + (0.379 + 0.275i)17-s + (0.115 + 0.354i)19-s + (−0.473 + 0.344i)20-s + (−0.650 + 0.444i)22-s + 0.168·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.838 - 0.544i$
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: \((2,\ 693,\ (\ :1/2),\ 0.838 - 0.544i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.32788 + 0.689892i\)
\(L(\frac12)\) \(\approx\) \(2.32788 + 0.689892i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.31 + 0.0938i)T \)
good2 \( 1 + (-0.901 + 0.655i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-2.79 - 2.03i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-1.66 + 1.21i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.56 - 1.13i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.501 - 1.54i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 0.807T + 23T^{2} \)
29 \( 1 + (2.46 - 7.58i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.637 - 0.463i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.10 + 9.56i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.657 - 2.02i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.08T + 43T^{2} \)
47 \( 1 + (2.33 + 7.19i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-8.75 + 6.36i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.01 + 3.13i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.871 - 0.632i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 2.40T + 67T^{2} \)
71 \( 1 + (2.57 + 1.87i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.378 + 1.16i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-7.67 + 5.57i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (13.0 + 9.44i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 4.43T + 89T^{2} \)
97 \( 1 + (5.23 - 3.80i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.67303144718135442265186790284, −10.00171672884846311370058916885, −8.867278999300839440536522806994, −7.85358171679322561178660444084, −7.01163089104611290338518039803, −5.77896175418489894776580653571, −5.20199150162017046779094742677, −3.77509667154851352221476875545, −2.90711731138263819772006868172, −1.92141272011380821852846426029, 1.17008457610744164994574902648, 2.52846524867477749231537838947, 4.27134444967573026120783386013, 5.17095311658345245944524979290, 5.70034069630998603119219155362, 6.45546201363085114889013765811, 7.70954557768239744289077695604, 8.789704194318368487047341261244, 9.598756484223977674737635820725, 10.11101061320047403825636951416

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