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Properties

Label 2-799-799.281-c0-0-4
Degree $2$
Conductor $799$
Sign $-0.979 + 0.202i$
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  + (1.34 − 1.34i)2-s + (−0.965 − 0.399i)3-s − 2.61i·4-s + (−1.83 + 0.760i)6-s + (0.0600 + 0.144i)7-s + (−2.17 − 2.17i)8-s + (0.0650 + 0.0650i)9-s + (−1.04 + 2.52i)12-s + (0.275 + 0.114i)14-s − 3.23·16-s + (0.809 + 0.587i)17-s + 0.175·18-s − 0.163i·21-s + (1.23 + 2.97i)24-s + (−0.707 − 0.707i)25-s + ⋯
L(s)  = 1  + (1.34 − 1.34i)2-s + (−0.965 − 0.399i)3-s − 2.61i·4-s + (−1.83 + 0.760i)6-s + (0.0600 + 0.144i)7-s + (−2.17 − 2.17i)8-s + (0.0650 + 0.0650i)9-s + (−1.04 + 2.52i)12-s + (0.275 + 0.114i)14-s − 3.23·16-s + (0.809 + 0.587i)17-s + 0.175·18-s − 0.163i·21-s + (1.23 + 2.97i)24-s + (−0.707 − 0.707i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $-0.979 + 0.202i$
Analytic conductor: \(0.398752\)
Root analytic conductor: \(0.631468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{799} (281, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 799,\ (\ :0),\ -0.979 + 0.202i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.341206277\)
\(L(\frac12)\) \(\approx\) \(1.341206277\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-0.809 - 0.587i)T \)
47 \( 1 + iT \)
good2 \( 1 + (-1.34 + 1.34i)T - iT^{2} \)
3 \( 1 + (0.965 + 0.399i)T + (0.707 + 0.707i)T^{2} \)
5 \( 1 + (0.707 + 0.707i)T^{2} \)
7 \( 1 + (-0.0600 - 0.144i)T + (-0.707 + 0.707i)T^{2} \)
11 \( 1 + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.707 + 0.707i)T^{2} \)
31 \( 1 + (-0.707 - 0.707i)T^{2} \)
37 \( 1 + (-1.57 - 0.652i)T + (0.707 + 0.707i)T^{2} \)
41 \( 1 + (0.707 - 0.707i)T^{2} \)
43 \( 1 - iT^{2} \)
53 \( 1 + (-0.642 + 0.642i)T - iT^{2} \)
59 \( 1 + (-0.437 - 0.437i)T + iT^{2} \)
61 \( 1 + (-0.581 - 1.40i)T + (-0.707 + 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (1.79 + 0.744i)T + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (0.707 + 0.707i)T^{2} \)
79 \( 1 + (1.79 - 0.744i)T + (0.707 - 0.707i)T^{2} \)
83 \( 1 + (1 - i)T - iT^{2} \)
89 \( 1 - 1.90iT - T^{2} \)
97 \( 1 + (-0.497 + 1.20i)T + (-0.707 - 0.707i)T^{2} \)
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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

−10.35092997999824597973500051509, −9.859736775048510603699295800498, −8.559606026577655924193663101755, −7.04293820468140210496215796913, −5.95897638871442382917198102240, −5.65480632060527779831597570545, −4.56822610504917698250411205351, −3.61626603399814769838508462041, −2.43978365373120494013713083807, −1.12032558219798655196954206314, 2.88335228674321957833288608823, 4.07980234028456895739692246494, 4.81461616176181746872550031763, 5.69683521035988789844062148963, 6.08404415854813004472361907119, 7.28138900562112459681604797370, 7.80077280664872740052851061414, 8.941413934642124297844309709739, 10.09217958243188944582337734021, 11.32135055946026197356191293527

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