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Properties

Label 2-693-77.10-c1-0-9
Degree $2$
Conductor $693$
Sign $0.292 - 0.956i$
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  + (1.49 − 0.863i)2-s + (0.489 − 0.848i)4-s + (−1.43 + 0.827i)5-s + (−2.00 + 1.72i)7-s + 1.76i·8-s + (−1.42 + 2.47i)10-s + (−3.00 + 1.40i)11-s + 2.97·13-s + (−1.51 + 4.30i)14-s + (2.49 + 4.32i)16-s + (−0.747 + 1.29i)17-s + (2.75 + 4.77i)19-s + 1.62i·20-s + (−3.27 + 4.69i)22-s + (−4.31 − 7.47i)23-s + ⋯
L(s)  = 1  + (1.05 − 0.610i)2-s + (0.244 − 0.424i)4-s + (−0.641 + 0.370i)5-s + (−0.758 + 0.651i)7-s + 0.622i·8-s + (−0.451 + 0.782i)10-s + (−0.905 + 0.424i)11-s + 0.825·13-s + (−0.404 + 1.15i)14-s + (0.624 + 1.08i)16-s + (−0.181 + 0.314i)17-s + (0.632 + 1.09i)19-s + 0.362i·20-s + (−0.698 + 1.00i)22-s + (−0.899 − 1.55i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.292 - 0.956i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ 0.292 - 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30714 + 0.966925i\)
\(L(\frac12)\) \(\approx\) \(1.30714 + 0.966925i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.00 - 1.72i)T \)
11 \( 1 + (3.00 - 1.40i)T \)
good2 \( 1 + (-1.49 + 0.863i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (1.43 - 0.827i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 2.97T + 13T^{2} \)
17 \( 1 + (0.747 - 1.29i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.75 - 4.77i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.31 + 7.47i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.02iT - 29T^{2} \)
31 \( 1 + (-7.67 - 4.43i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.11 + 7.12i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.12T + 41T^{2} \)
43 \( 1 + 5.03iT - 43T^{2} \)
47 \( 1 + (-6.38 + 3.68i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.630 - 1.09i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.30 - 2.48i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.82 - 3.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.36 + 9.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.05T + 71T^{2} \)
73 \( 1 + (4.92 - 8.53i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.41 - 1.96i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 + (-4.47 + 2.58i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.35iT - 97T^{2} \)
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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

−10.64130343574732721990392089434, −10.26184905447757660484327258891, −8.761282672127380989717785275885, −8.159990315515086823747048851515, −6.95816849463744232442964156619, −5.90737923411179129929975256775, −5.09125694341782220006328360678, −3.86816662420146567909365105584, −3.23973619110682212491272306717, −2.14528479343488797281832811092, 0.59974873741088830672775833334, 3.04180655048923348116774679404, 3.92652640047826257389720405596, 4.75185277472398445217913164452, 5.78957838876574812131826086504, 6.54058741007725733308623023903, 7.53629266816715979297886852762, 8.230247894995192704284935719961, 9.573432083484373922740885733681, 10.17399586614127959220506997023

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