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Properties

Label 2-798-399.107-c1-0-33
Degree $2$
Conductor $798$
Sign $0.999 - 0.00176i$
Analytic cond. $6.37206$
Root an. cond. $2.52429$
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  + 2-s + (1.25 − 1.19i)3-s + 4-s + 1.95i·5-s + (1.25 − 1.19i)6-s + (1.98 + 1.74i)7-s + 8-s + (0.129 − 2.99i)9-s + 1.95i·10-s + (−1.30 + 0.750i)11-s + (1.25 − 1.19i)12-s + (−0.271 − 0.156i)13-s + (1.98 + 1.74i)14-s + (2.34 + 2.44i)15-s + 16-s + (0.475 + 0.274i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.722 − 0.691i)3-s + 0.5·4-s + 0.875i·5-s + (0.510 − 0.489i)6-s + (0.751 + 0.659i)7-s + 0.353·8-s + (0.0430 − 0.999i)9-s + 0.618i·10-s + (−0.392 + 0.226i)11-s + (0.361 − 0.345i)12-s + (−0.0752 − 0.0434i)13-s + (0.531 + 0.466i)14-s + (0.605 + 0.631i)15-s + 0.250·16-s + (0.115 + 0.0666i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00176i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.999 - 0.00176i$
Analytic conductor: \(6.37206\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{798} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 798,\ (\ :1/2),\ 0.999 - 0.00176i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.18048 + 0.00281003i\)
\(L(\frac12)\) \(\approx\) \(3.18048 + 0.00281003i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-1.25 + 1.19i)T \)
7 \( 1 + (-1.98 - 1.74i)T \)
19 \( 1 + (-4.35 - 0.0337i)T \)
good5 \( 1 - 1.95iT - 5T^{2} \)
11 \( 1 + (1.30 - 0.750i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.271 + 0.156i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.475 - 0.274i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.91 - 1.68i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.14 + 1.98i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.673 + 0.388i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.22 + 1.28i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.41 + 4.17i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.59 + 4.50i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (8.03 - 4.63i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 + (-5.30 + 9.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.26 - 9.12i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + 0.0196iT - 67T^{2} \)
71 \( 1 + (3.62 + 6.28i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.75 + 6.49i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 5.37iT - 79T^{2} \)
83 \( 1 + 11.1iT - 83T^{2} \)
89 \( 1 + (2.48 + 4.30i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.00 - 1.15i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.34350529155517744202077112674, −9.407275308038376287295233404907, −8.284463199421439463978472451716, −7.63263981141253183118309943422, −6.84515385563370732906937393836, −5.92000947904744624821317579018, −4.92945183339306881770464052067, −3.55233302866369676555541567372, −2.71083753960450670926779659222, −1.75271849379960665196936416981, 1.48326516271910058160201566245, 2.93296969465563985836490093849, 3.97418188584524573852404507310, 4.87146859456714510933818043092, 5.31133868920336538876852471861, 6.84274090164851345495637521120, 7.973631088741487586722025556220, 8.349664645731497928100798277370, 9.507830331468551617811794193546, 10.25517334264926653485318063391

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