Properties

Label 2-798-7.2-c1-0-12
Degree $2$
Conductor $798$
Sign $0.997 - 0.0669i$
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  + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.574 − 0.995i)5-s + 0.999·6-s + (0.00953 + 2.64i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.574 − 0.995i)10-s + (1.08 + 1.87i)11-s + (0.499 − 0.866i)12-s + 2.29·13-s + (2.29 + 1.31i)14-s + 1.14·15-s + (−0.5 + 0.866i)16-s + (2.49 + 4.32i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.256 − 0.445i)5-s + 0.408·6-s + (0.00360 + 0.999i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.181 − 0.314i)10-s + (0.326 + 0.566i)11-s + (0.144 − 0.249i)12-s + 0.637·13-s + (0.613 + 0.351i)14-s + 0.296·15-s + (−0.125 + 0.216i)16-s + (0.605 + 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.997 - 0.0669i$
Analytic conductor: \(6.37206\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{798} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 798,\ (\ :1/2),\ 0.997 - 0.0669i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13865 + 0.0716478i\)
\(L(\frac12)\) \(\approx\) \(2.13865 + 0.0716478i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.00953 - 2.64i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.574 + 0.995i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.08 - 1.87i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.29T + 13T^{2} \)
17 \( 1 + (-2.49 - 4.32i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3.45 + 5.98i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.76T + 29T^{2} \)
31 \( 1 + (2.19 + 3.79i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.330 - 0.571i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.806T + 41T^{2} \)
43 \( 1 + 2.08T + 43T^{2} \)
47 \( 1 + (-1.86 + 3.22i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.63 - 9.76i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.21 + 9.02i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.28 - 2.23i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.44 + 7.69i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.323T + 71T^{2} \)
73 \( 1 + (-2.36 - 4.09i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.79 + 6.57i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.17T + 83T^{2} \)
89 \( 1 + (6.18 - 10.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.91T + 97T^{2} \)
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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

−10.34900793728620015164863704298, −9.380693645929766825081584337565, −8.857010486784435388930062963152, −8.058627646768166122300308215806, −6.51693891244378325326398111652, −5.64393211440478958116883490432, −4.79164347947605254552761939162, −3.81887054726243178533368640560, −2.70564315008875863096012538597, −1.53698439465977553823294236674, 1.08349893738728165568500955602, 2.92650565585940772943296461021, 3.73073809940891471608590697294, 4.98539480678782786781165719128, 6.05654631980702478730694403137, 6.90047704487871449249591695451, 7.42470296640351382766537952136, 8.400973121428156839132043799077, 9.245009076314920878474391180848, 10.24713341153564759886462916487

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