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Properties

Label 2-693-7.2-c1-0-11
Degree $2$
Conductor $693$
Sign $0.118 - 0.992i$
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  + (−0.276 + 0.478i)2-s + (0.847 + 1.46i)4-s + (0.795 − 1.37i)5-s + (0.886 + 2.49i)7-s − 2.04·8-s + (0.439 + 0.760i)10-s + (−0.5 − 0.866i)11-s + 2.87·13-s + (−1.43 − 0.264i)14-s + (−1.13 + 1.95i)16-s + (2.41 + 4.18i)17-s + (0.572 − 0.992i)19-s + 2.69·20-s + 0.552·22-s + (−1.82 + 3.16i)23-s + ⋯
L(s)  = 1  + (−0.195 + 0.338i)2-s + (0.423 + 0.733i)4-s + (0.355 − 0.615i)5-s + (0.335 + 0.942i)7-s − 0.721·8-s + (0.138 + 0.240i)10-s + (−0.150 − 0.261i)11-s + 0.798·13-s + (−0.384 − 0.0706i)14-s + (−0.282 + 0.489i)16-s + (0.586 + 1.01i)17-s + (0.131 − 0.227i)19-s + 0.602·20-s + 0.117·22-s + (−0.380 + 0.659i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.118 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.118 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.118 - 0.992i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ 0.118 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20362 + 1.06888i\)
\(L(\frac12)\) \(\approx\) \(1.20362 + 1.06888i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.886 - 2.49i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.276 - 0.478i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.795 + 1.37i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 2.87T + 13T^{2} \)
17 \( 1 + (-2.41 - 4.18i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.572 + 0.992i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.82 - 3.16i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.325T + 29T^{2} \)
31 \( 1 + (3.22 + 5.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.847 - 1.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.05T + 41T^{2} \)
43 \( 1 - 4.62T + 43T^{2} \)
47 \( 1 + (-0.152 + 0.264i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.85 - 4.94i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.93 - 10.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.886 - 1.53i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.63 + 13.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.16T + 71T^{2} \)
73 \( 1 + (-5.86 - 10.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.35 + 7.54i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.40T + 83T^{2} \)
89 \( 1 + (2.87 - 4.97i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.65T + 97T^{2} \)
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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

−10.79846967185766954230255160259, −9.484778834714513355720155549284, −8.783988168961956592613725771907, −8.180115812866724977833067417966, −7.33277799332082149020284272925, −5.97717821933176128086367905696, −5.64625144953556547059251116268, −4.13188685754835853126872866666, −2.97729711060326083206686738474, −1.67594575944611683732055197894, 0.971547334627725741580565518022, 2.28225935909294695506267709265, 3.46003492026133032423989807151, 4.81920563112858470432290573431, 5.87468511616766347349932527378, 6.75953848884527165825971240332, 7.46541547881080104872506290985, 8.668928592332088228162735211389, 9.746920777047776826610838528506, 10.35337698482755819994941740305

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