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Properties

Label 2-693-7.4-c1-0-13
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: $\chi_{693} (298, \cdot )$
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

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

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