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Properties

Label 2-799-799.140-c0-0-3
Degree $2$
Conductor $799$
Sign $0.559 - 0.828i$
Analytic cond. $0.398752$
Root an. cond. $0.631468$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.61i·2-s + (1.26 − 1.26i)3-s − 1.61·4-s + (2.03 + 2.03i)6-s + (0.221 + 0.221i)7-s − i·8-s − 2.17i·9-s + (−2.03 + 2.03i)12-s + (−0.357 + 0.357i)14-s + (0.309 + 0.951i)17-s + 3.52·18-s + 0.557·21-s + (−1.26 − 1.26i)24-s − i·25-s + (−1.48 − 1.48i)27-s + (−0.357 − 0.357i)28-s + ⋯
L(s)  = 1  + 1.61i·2-s + (1.26 − 1.26i)3-s − 1.61·4-s + (2.03 + 2.03i)6-s + (0.221 + 0.221i)7-s − i·8-s − 2.17i·9-s + (−2.03 + 2.03i)12-s + (−0.357 + 0.357i)14-s + (0.309 + 0.951i)17-s + 3.52·18-s + 0.557·21-s + (−1.26 − 1.26i)24-s − i·25-s + (−1.48 − 1.48i)27-s + (−0.357 − 0.357i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $0.559 - 0.828i$
Analytic conductor: \(0.398752\)
Root analytic conductor: \(0.631468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{799} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 799,\ (\ :0),\ 0.559 - 0.828i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.346132447\)
\(L(\frac12)\) \(\approx\) \(1.346132447\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-0.309 - 0.951i)T \)
47 \( 1 + T \)
good2 \( 1 - 1.61iT - T^{2} \)
3 \( 1 + (-1.26 + 1.26i)T - iT^{2} \)
5 \( 1 + iT^{2} \)
7 \( 1 + (-0.221 - 0.221i)T + iT^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (1.26 - 1.26i)T - iT^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + T^{2} \)
53 \( 1 - 1.17iT - T^{2} \)
59 \( 1 - 1.61iT - T^{2} \)
61 \( 1 + (1.39 + 1.39i)T + iT^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.642 - 0.642i)T - iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + (0.642 + 0.642i)T + iT^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 1.61T + T^{2} \)
97 \( 1 + (-1.39 + 1.39i)T - iT^{2} \)
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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

−10.23710192302465354025523798128, −9.118585044841358588413282848099, −8.436732208733820339703100159372, −8.051040635193276788544333920243, −7.21436611902793590952466696829, −6.53042044374655625098902584477, −5.77088800365067881061283809805, −4.44045143367586847739925641252, −3.11060227590902797202070711377, −1.74748537697137233320452185668, 1.83704402982401397509200382824, 2.95224292272376695219224146868, 3.56346144148815042117010944740, 4.46247090255594849826074991755, 5.24966075159384666072122074519, 7.24522557810320702214811280806, 8.279084303159274878092212217250, 9.167478530373659456992748633535, 9.542142770875046362003247528899, 10.35288368516807251489070915387

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