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Properties

Label 2-799-799.234-c0-0-4
Degree $2$
Conductor $799$
Sign $-0.939 - 0.341i$
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 + (−0.642 − 0.642i)3-s − 1.61·4-s + (−1.03 + 1.03i)6-s + (1.39 − 1.39i)7-s + i·8-s − 0.175i·9-s + (1.03 + 1.03i)12-s + (−2.26 − 2.26i)14-s + (0.309 + 0.951i)17-s − 0.284·18-s − 1.79·21-s + (0.642 − 0.642i)24-s + i·25-s + (−0.754 + 0.754i)27-s + (−2.26 + 2.26i)28-s + ⋯
L(s)  = 1  − 1.61i·2-s + (−0.642 − 0.642i)3-s − 1.61·4-s + (−1.03 + 1.03i)6-s + (1.39 − 1.39i)7-s + i·8-s − 0.175i·9-s + (1.03 + 1.03i)12-s + (−2.26 − 2.26i)14-s + (0.309 + 0.951i)17-s − 0.284·18-s − 1.79·21-s + (0.642 − 0.642i)24-s + i·25-s + (−0.754 + 0.754i)27-s + (−2.26 + 2.26i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $-0.939 - 0.341i$
Analytic conductor: \(0.398752\)
Root analytic conductor: \(0.631468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{799} (234, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 799,\ (\ :0),\ -0.939 - 0.341i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8282538186\)
\(L(\frac12)\) \(\approx\) \(0.8282538186\)
\(L(1)\) 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 \( 1 + (0.642 + 0.642i)T + iT^{2} \)
5 \( 1 - iT^{2} \)
7 \( 1 + (-1.39 + 1.39i)T - iT^{2} \)
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 \( 1 + (-0.642 - 0.642i)T + iT^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + T^{2} \)
53 \( 1 - 1.17iT - T^{2} \)
59 \( 1 + 1.61iT - T^{2} \)
61 \( 1 + (0.221 - 0.221i)T - iT^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-1.26 - 1.26i)T + iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (-1.26 + 1.26i)T - iT^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 1.61T + T^{2} \)
97 \( 1 + (-0.221 - 0.221i)T + iT^{2} \)
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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

−10.36276273234322013119571749242, −9.546023718494355919555346824137, −8.356620478294143936276681588118, −7.52461668341903366279125653933, −6.56700279282551973566623634428, −5.22754030500683862527808367035, −4.27425796892834831771647267274, −3.45622213338103050020346655998, −1.76000343888954358364343851002, −1.06632027335765596283969352211, 2.33726233986095704338060907860, 4.39969481491432142859437293153, 5.08973779805699532604407337442, 5.53987265476286844157492164347, 6.40923652133995615017580660110, 7.64335380440630616265623353621, 8.195149374214521307408596146706, 8.977167894973261517257512384732, 9.828090828557289385108406115400, 11.06532285430356677351376511829

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