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Properties

Label 2-798-133.12-c1-0-4
Degree $2$
Conductor $798$
Sign $0.792 - 0.609i$
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  − i·2-s + (0.5 − 0.866i)3-s − 4-s + 3.03i·5-s + (−0.866 − 0.5i)6-s + (−1.22 − 2.34i)7-s + i·8-s + (−0.499 − 0.866i)9-s + 3.03·10-s + (2.50 + 4.33i)11-s + (−0.5 + 0.866i)12-s + (−2.77 + 4.81i)13-s + (−2.34 + 1.22i)14-s + (2.63 + 1.51i)15-s + 16-s + (−0.111 − 0.0643i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.288 − 0.499i)3-s − 0.5·4-s + 1.35i·5-s + (−0.353 − 0.204i)6-s + (−0.464 − 0.885i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + 0.960·10-s + (0.754 + 1.30i)11-s + (−0.144 + 0.249i)12-s + (−0.770 + 1.33i)13-s + (−0.626 + 0.328i)14-s + (0.679 + 0.392i)15-s + 0.250·16-s + (−0.0270 − 0.0156i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.792 - 0.609i$
Analytic conductor: \(6.37206\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{798} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 798,\ (\ :1/2),\ 0.792 - 0.609i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14521 + 0.389656i\)
\(L(\frac12)\) \(\approx\) \(1.14521 + 0.389656i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (1.22 + 2.34i)T \)
19 \( 1 + (4.32 + 0.547i)T \)
good5 \( 1 - 3.03iT - 5T^{2} \)
11 \( 1 + (-2.50 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.77 - 4.81i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.111 + 0.0643i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3.20 - 5.55i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.02 + 1.16i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.58 - 4.47i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.73 - 2.73i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.12 + 5.41i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.02 - 3.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.78 + 2.18i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 3.60iT - 53T^{2} \)
59 \( 1 + (3.99 - 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.5 + 6.06i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 - 9.12iT - 67T^{2} \)
71 \( 1 + (-6.64 + 3.83i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (10.3 + 5.99i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 9.08iT - 79T^{2} \)
83 \( 1 + 15.7iT - 83T^{2} \)
89 \( 1 + (3.76 + 6.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.70 - 4.67i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.29683601388644018302613206628, −9.713078847379228918491180822981, −8.942825047710060290328307651579, −7.40298315846891616376718851634, −7.06688393221605897962661809020, −6.35476911997203552312190895219, −4.56836911119414729888281839408, −3.77202286018595079133571601845, −2.66706962173452832752139407059, −1.67726272149285162336979849737, 0.58916559083521614716320096266, 2.69972924779184387761205585758, 3.94292345703009256270836333916, 4.97973359530985878155598371172, 5.66555781650342978039346220270, 6.47575754133096044137940586314, 8.004119867806082624806371873254, 8.533689883944503337636977491414, 9.079570502755053509809083863275, 9.825200067235167161898938603444

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