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Properties

Label 2-798-399.107-c1-0-18
Degree $2$
Conductor $798$
Sign $0.997 - 0.0654i$
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 + (−0.423 − 1.67i)3-s + 4-s + 1.80i·5-s + (−0.423 − 1.67i)6-s + (−2.02 + 1.70i)7-s + 8-s + (−2.64 + 1.42i)9-s + 1.80i·10-s + (2.22 − 1.28i)11-s + (−0.423 − 1.67i)12-s + (4.31 + 2.49i)13-s + (−2.02 + 1.70i)14-s + (3.03 − 0.764i)15-s + 16-s + (2.80 + 1.61i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.244 − 0.969i)3-s + 0.5·4-s + 0.807i·5-s + (−0.172 − 0.685i)6-s + (−0.764 + 0.644i)7-s + 0.353·8-s + (−0.880 + 0.474i)9-s + 0.571i·10-s + (0.672 − 0.387i)11-s + (−0.122 − 0.484i)12-s + (1.19 + 0.690i)13-s + (−0.540 + 0.455i)14-s + (0.783 − 0.197i)15-s + 0.250·16-s + (0.680 + 0.392i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.997 - 0.0654i$
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: \((2,\ 798,\ (\ :1/2),\ 0.997 - 0.0654i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.18274 + 0.0714604i\)
\(L(\frac12)\) \(\approx\) \(2.18274 + 0.0714604i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.423 + 1.67i)T \)
7 \( 1 + (2.02 - 1.70i)T \)
19 \( 1 + (-3.26 + 2.88i)T \)
good5 \( 1 - 1.80iT - 5T^{2} \)
11 \( 1 + (-2.22 + 1.28i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.31 - 2.49i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.80 - 1.61i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-5.53 + 3.19i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.00 - 5.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.30 - 0.754i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.45 + 3.72i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.96 - 5.13i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.581 + 1.00i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (8.27 - 4.77i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.41T + 53T^{2} \)
59 \( 1 + (3.43 - 5.95i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.243 - 0.421i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + 14.8iT - 67T^{2} \)
71 \( 1 + (-5.40 - 9.35i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.40 - 4.15i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 9.15iT - 79T^{2} \)
83 \( 1 - 8.99iT - 83T^{2} \)
89 \( 1 + (3.01 + 5.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-14.3 + 8.26i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.73737234718615109999719590081, −9.276786330778421843044677653322, −8.571693069367142177396897775034, −7.28003902940001645111023942898, −6.61358984570376042611193974288, −6.13298834319408585402217718856, −5.15231845933533473001610551905, −3.49263064660151609750693105628, −2.88848254744245129724016034032, −1.42776133646379044891418322424, 1.07706520692210756995328468482, 3.32249676465241171544419966338, 3.73800227355775598903444166385, 4.88513761432296786795959576484, 5.60478147651421715834569540799, 6.49606469753972666638519860639, 7.61881977797046083557646232099, 8.760830331321247197292889837640, 9.567776845076215125157628797261, 10.23354140152748156009158949238

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