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Properties

Label 2-693-7.2-c1-0-29
Degree $2$
Conductor $693$
Sign $-0.363 + 0.931i$
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  + (0.643 − 1.11i)2-s + (0.171 + 0.296i)4-s + (1.95 − 3.39i)5-s + (−0.234 − 2.63i)7-s + 3.01·8-s + (−2.52 − 4.36i)10-s + (−0.5 − 0.866i)11-s − 3.04·13-s + (−3.08 − 1.43i)14-s + (1.59 − 2.76i)16-s + (1.98 + 3.44i)17-s + (−3.79 + 6.57i)19-s + 1.34·20-s − 1.28·22-s + (2.25 − 3.90i)23-s + ⋯
L(s)  = 1  + (0.455 − 0.788i)2-s + (0.0856 + 0.148i)4-s + (0.875 − 1.51i)5-s + (−0.0885 − 0.996i)7-s + 1.06·8-s + (−0.797 − 1.38i)10-s + (−0.150 − 0.261i)11-s − 0.843·13-s + (−0.825 − 0.383i)14-s + (0.399 − 0.692i)16-s + (0.481 + 0.834i)17-s + (−0.870 + 1.50i)19-s + 0.300·20-s − 0.274·22-s + (0.470 − 0.814i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.363 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.363 + 0.931i$
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: \((2,\ 693,\ (\ :1/2),\ -0.363 + 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31672 - 1.92629i\)
\(L(\frac12)\) \(\approx\) \(1.31672 - 1.92629i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.234 + 2.63i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.643 + 1.11i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.95 + 3.39i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 3.04T + 13T^{2} \)
17 \( 1 + (-1.98 - 3.44i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.79 - 6.57i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.25 + 3.90i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.75T + 29T^{2} \)
31 \( 1 + (-3.37 - 5.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.171 - 0.296i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.79T + 41T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 + (-0.828 + 1.43i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.47 + 11.2i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.83 + 3.17i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.234 + 0.405i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.28 - 2.23i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.00T + 71T^{2} \)
73 \( 1 + (-4.36 - 7.56i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.359 - 0.623i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.5T + 83T^{2} \)
89 \( 1 + (6.17 - 10.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.1T + 97T^{2} \)
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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

−10.29130827137076573985758302154, −9.616952575274705085160702605056, −8.415071190788262680431273893222, −7.81621663085720393710227774795, −6.52569400836260047459698561555, −5.37522559812168880967676387211, −4.50770376737198839066963067656, −3.70788911283629576013094729562, −2.18272001752226645472511829522, −1.14846418186909967931224067904, 2.19891043181301778298131367874, 2.84509937878824010384602672624, 4.65155197149474738079474822359, 5.62051547410178261216929974668, 6.20520920536163684674553283140, 7.10153634117146000614316306224, 7.60289699352125996460944707735, 9.288811842874564832116273688616, 9.734181058138933444048946276068, 10.82751253670234292246231024817

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