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Properties

Label 2-693-11.3-c1-0-3
Degree $2$
Conductor $693$
Sign $-0.908 - 0.417i$
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  + (0.5 − 0.363i)2-s + (−0.5 + 1.53i)4-s + (−1.30 − 0.951i)5-s + (−0.309 + 0.951i)7-s + (0.690 + 2.12i)8-s − 10-s + (−1.69 − 2.85i)11-s + (−3.42 + 2.48i)13-s + (0.190 + 0.587i)14-s + (−1.49 − 1.08i)16-s + (−2.80 − 2.04i)17-s + (2 + 6.15i)19-s + (2.11 − 1.53i)20-s + (−1.88 − 0.812i)22-s − 5.70·23-s + ⋯
L(s)  = 1  + (0.353 − 0.256i)2-s + (−0.250 + 0.769i)4-s + (−0.585 − 0.425i)5-s + (−0.116 + 0.359i)7-s + (0.244 + 0.751i)8-s − 0.316·10-s + (−0.509 − 0.860i)11-s + (−0.950 + 0.690i)13-s + (0.0510 + 0.157i)14-s + (−0.374 − 0.272i)16-s + (−0.681 − 0.494i)17-s + (0.458 + 1.41i)19-s + (0.473 − 0.344i)20-s + (−0.401 − 0.173i)22-s − 1.19·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 - 0.417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.908 - 0.417i$
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: \((2,\ 693,\ (\ :1/2),\ -0.908 - 0.417i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0855030 + 0.390379i\)
\(L(\frac12)\) \(\approx\) \(0.0855030 + 0.390379i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 + 2.85i)T \)
good2 \( 1 + (-0.5 + 0.363i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (1.30 + 0.951i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (3.42 - 2.48i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.80 + 2.04i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2 - 6.15i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 5.70T + 23T^{2} \)
29 \( 1 + (-0.0729 + 0.224i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.42 - 1.76i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.85 + 5.70i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.04 - 9.37i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 11.4T + 43T^{2} \)
47 \( 1 + (-1.59 - 4.89i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (10.1 - 7.38i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.42 + 4.39i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-6.42 - 4.66i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 13.2T + 67T^{2} \)
71 \( 1 + (3.66 + 2.66i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.28 + 7.02i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.92 + 3.57i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-6.85 - 4.97i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 1.32T + 89T^{2} \)
97 \( 1 + (-4.28 + 3.11i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.14494909539248477881214212928, −9.900458533636488753103580390146, −9.032217964177226318335183594165, −8.084594686000080790456746832899, −7.70668899945549479705319837961, −6.33379960139674838847760322056, −5.17011416139620924691766659071, −4.31532443791230400248343365722, −3.37941293515540962600927196204, −2.22781243826551344460616110327, 0.17561500791405409368380524516, 2.18081934996258749744625965102, 3.65119795049187197927746645661, 4.69642452675996519598924819967, 5.39027684499238006843559045330, 6.69199419228807433639188765091, 7.23217141924790180938536399097, 8.190144885847866294919088542680, 9.509059901026695474734414769618, 10.06210261200593961687725364384

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