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Properties

Label 2-693-7.4-c1-0-21
Degree $2$
Conductor $693$
Sign $0.714 - 0.699i$
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  + (1.17 + 2.03i)2-s + (−1.76 + 3.06i)4-s + (−2.05 − 3.56i)5-s + (2.51 − 0.818i)7-s − 3.60·8-s + (4.84 − 8.38i)10-s + (0.5 − 0.866i)11-s + 5.39·13-s + (4.62 + 4.16i)14-s + (−0.709 − 1.22i)16-s + (1.17 − 2.03i)17-s + (−2.87 − 4.97i)19-s + 14.5·20-s + 2.35·22-s + (0.709 + 1.22i)23-s + ⋯
L(s)  = 1  + (0.831 + 1.44i)2-s + (−0.883 + 1.53i)4-s + (−0.920 − 1.59i)5-s + (0.950 − 0.309i)7-s − 1.27·8-s + (1.53 − 2.65i)10-s + (0.150 − 0.261i)11-s + 1.49·13-s + (1.23 + 1.11i)14-s + (−0.177 − 0.307i)16-s + (0.285 − 0.494i)17-s + (−0.659 − 1.14i)19-s + 3.25·20-s + 0.501·22-s + (0.147 + 0.256i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.714 - 0.699i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ 0.714 - 0.699i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04334 + 0.833825i\)
\(L(\frac12)\) \(\approx\) \(2.04334 + 0.833825i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.51 + 0.818i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-1.17 - 2.03i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (2.05 + 3.56i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 5.39T + 13T^{2} \)
17 \( 1 + (-1.17 + 2.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.87 + 4.97i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.709 - 1.22i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.13T + 29T^{2} \)
31 \( 1 + (-0.266 + 0.461i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.20 + 2.08i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.68T + 41T^{2} \)
43 \( 1 - 3.42T + 43T^{2} \)
47 \( 1 + (-4.47 - 7.74i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.80 + 3.12i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.709 + 1.22i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.88 + 3.25i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.01 - 1.75i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.10T + 71T^{2} \)
73 \( 1 + (3.23 - 5.60i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.09 + 8.82i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.0T + 83T^{2} \)
89 \( 1 + (4.36 + 7.56i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 17.7T + 97T^{2} \)
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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

−10.83828350658022471531108631422, −9.022565330798251160962046742532, −8.534573978467274623047032352379, −7.947610774000654510263632812256, −7.09589294175325330684298271320, −5.95723741900775980410055617407, −5.01449760995139523208209285953, −4.47838814065459067695367429844, −3.67667332530252493710923534293, −1.07470225652318955890028867875, 1.58433340439621669089557031455, 2.76001976832178306171493710168, 3.72413205719258396810168783212, 4.27180071620012993746122109813, 5.67876837533392274400931028504, 6.64083790264329158287213474591, 7.86016431622353679871204958093, 8.623605519789850417720091093123, 10.23716218363772511721415284905, 10.60265694207657116737653064634

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