↑

Properties

Label 2-798-399.107-c1-0-37
Degree $2$
Conductor $798$
Sign $0.333 + 0.942i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + (−1.57 − 0.730i)3-s + 4-s − 2.50i·5-s + (−1.57 − 0.730i)6-s + (2.58 + 0.563i)7-s + 8-s + (1.93 + 2.29i)9-s − 2.50i·10-s + (1.03 − 0.597i)11-s + (−1.57 − 0.730i)12-s + (1.93 + 1.11i)13-s + (2.58 + 0.563i)14-s + (−1.82 + 3.92i)15-s + 16-s + (−1.80 − 1.03i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.906 − 0.422i)3-s + 0.5·4-s − 1.11i·5-s + (−0.641 − 0.298i)6-s + (0.977 + 0.212i)7-s + 0.353·8-s + (0.643 + 0.765i)9-s − 0.790i·10-s + (0.311 − 0.180i)11-s + (−0.453 − 0.211i)12-s + (0.537 + 0.310i)13-s + (0.690 + 0.150i)14-s + (−0.471 + 1.01i)15-s + 0.250·16-s + (−0.436 − 0.252i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.333 + 0.942i$
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: \((2,\ 798,\ (\ :1/2),\ 0.333 + 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64432 - 1.16209i\)
\(L(\frac12)\) \(\approx\) \(1.64432 - 1.16209i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.57 + 0.730i)T \)
7 \( 1 + (-2.58 - 0.563i)T \)
19 \( 1 + (3.72 + 2.26i)T \)
good5 \( 1 + 2.50iT - 5T^{2} \)
11 \( 1 + (-1.03 + 0.597i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.93 - 1.11i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.80 + 1.03i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-7.75 + 4.47i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.33 - 2.30i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.48 - 1.43i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.47 + 2.58i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.33 + 7.50i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.312 + 0.541i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.63 + 2.10i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.82T + 53T^{2} \)
59 \( 1 + (-5.03 + 8.72i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.74 - 9.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + 6.31iT - 67T^{2} \)
71 \( 1 + (0.267 + 0.462i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.50 - 11.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 12.8iT - 79T^{2} \)
83 \( 1 + 4.90iT - 83T^{2} \)
89 \( 1 + (-1.75 - 3.03i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.93 + 1.11i)T + (48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.58078875347482016845824631313, −8.926325806592498567327657586835, −8.545668888247001109126724551241, −7.22726176564016358067782197727, −6.53505784744967762494060580959, −5.33337762716597433993842197653, −4.95508504617608869175649902510, −4.05717621151312205074078520541, −2.14805504658069968886542996237, −1.01427495593248227200002910629, 1.60338839031368545596698679758, 3.21316335665987371766042278537, 4.15729598376892450004983448326, 5.05310991080260811343432709748, 6.00140872203800963207424668400, 6.76742591406533140078910351761, 7.50645398731857605119485664734, 8.743079055291083681029867090746, 10.00643530381578058848406838285, 10.78183680065809991299644622995

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